Check certain columns' values when using groupby in Pandas

I have a dataframe like this

df = pd.DataFrame({'Name': ['Bob', 'Bob', 'Bob', 'Joe', 'Joe', 'Joe'],
                'ID': [1,2,3,4,5,6],
                'Value': [1,1,1,0,0,1]})
df

 Name    ID    Value   
 Bob     1       1          
 Bob     2       1          
 Bob     3       1          
 Joe     4       0          
 Joe     5       0          
 Joe     6       1          

The goal is to compute a result column. This is done by checking each group in the name column, in this case Bob & Joe.

So for each group, if the values in the value column are all 1, the values in the result column for that group will be all 1. If the values are all 0, the result column values for that group will be all 0. And if the values are a mix of 1 and 0, the result column for that group will be all 0.

So the output should look like this:

Name    ID    Value    Result
 Bob     1       1       1   
 Bob     2       1       1   
 Bob     3       1       1   
 Joe     4       0       0   
 Joe     5       0       0   
 Joe     6       1       0   

The difficulty is creating these groups and then checking each one.

My attempt:

df = df.groupby('Name')

df['Result'] = df.apply(lambda x: x['Value'])

Use all with groupby+transform:

df['Result'] = df.groupby('Name')['Value'].transform('all').astype(int)
# or df['Result'] = df['Value'].eq(1).groupby(df['Name']).transform('all').astype(int)
print(df)

  Name  ID  Value  Result
0  Bob   1      1       1
1  Bob   2      1       1
2  Bob   3      1       1
3  Joe   4      0       0
4  Joe   5      0       0
5  Joe   6      1       0

IIUC

df['Result']=df.groupby('Name').Value.all().reindex(df.Name).astype(int).values
df
Out[57]: 
  Name  ID  Value  Result
0  Bob   1      1       1
1  Bob   2      1       1
2  Bob   3      1       1
3  Joe   4      0       0
4  Joe   5      0       0
5  Joe   6      1       0