Metamath Proof Explorer


Theorem ablodivdiv

Description: Law for double group division. (Contributed by NM, 29-Feb-2008) (New usage is discouraged.)

Ref Expression
Hypotheses abldiv.1 X=ranG
abldiv.3 D=/gG
Assertion ablodivdiv GAbelOpAXBXCXADBDC=ADBGC

Proof

Step Hyp Ref Expression
1 abldiv.1 X=ranG
2 abldiv.3 D=/gG
3 ablogrpo GAbelOpGGrpOp
4 1 2 grpodivdiv GGrpOpAXBXCXADBDC=AGCDB
5 3 4 sylan GAbelOpAXBXCXADBDC=AGCDB
6 3ancomb AXBXCXAXCXBX
7 1 2 grpomuldivass GGrpOpAXCXBXAGCDB=AGCDB
8 3 7 sylan GAbelOpAXCXBXAGCDB=AGCDB
9 1 2 ablomuldiv GAbelOpAXCXBXAGCDB=ADBGC
10 8 9 eqtr3d GAbelOpAXCXBXAGCDB=ADBGC
11 6 10 sylan2b GAbelOpAXBXCXAGCDB=ADBGC
12 5 11 eqtrd GAbelOpAXBXCXADBDC=ADBGC