Theorem gimco 17710

Description: The composition of group isomorphisms is a group isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)

Assertion

Ref	Expression
gimco	|- ( ( F e. ( T GrpIso U ) /\ G e. ( S GrpIso T ) ) -> ( F o. G ) e. ( S GrpIso U ) )

Proof of Theorem gimco

Step	Hyp	Ref	Expression
1		isgim2 17707	. . 3 |- ( F e. ( T GrpIso U ) <-> ( F e. ( T GrpHom U ) /\ `' F e. ( U GrpHom T ) ) )
2		isgim2 17707	. . 3 |- ( G e. ( S GrpIso T ) <-> ( G e. ( S GrpHom T ) /\ `' G e. ( T GrpHom S ) ) )
3		ghmco 17680	. . . . 5 |- ( ( F e. ( T GrpHom U ) /\ G e. ( S GrpHom T ) ) -> ( F o. G ) e. ( S GrpHom U ) )
4		cnvco 5308	. . . . . 6 |- `' ( F o. G ) = ( `' G o. `' F )
5		ghmco 17680	. . . . . . 7 |- ( ( `' G e. ( T GrpHom S ) /\ `' F e. ( U GrpHom T ) ) -> ( `' G o. `' F ) e. ( U GrpHom S ) )
6	5	ancoms 469	. . . . . 6 |- ( ( `' F e. ( U GrpHom T ) /\ `' G e. ( T GrpHom S ) ) -> ( `' G o. `' F ) e. ( U GrpHom S ) )
7	4, 6	syl5eqel 2705	. . . . 5 |- ( ( `' F e. ( U GrpHom T ) /\ `' G e. ( T GrpHom S ) ) -> `' ( F o. G ) e. ( U GrpHom S ) )
8	3, 7	anim12i 590	. . . 4 |- ( ( ( F e. ( T GrpHom U ) /\ G e. ( S GrpHom T ) ) /\ ( `' F e. ( U GrpHom T ) /\ `' G e. ( T GrpHom S ) ) ) -> ( ( F o. G ) e. ( S GrpHom U ) /\ `' ( F o. G ) e. ( U GrpHom S ) ) )
9	8	an4s 869	. . 3 |- ( ( ( F e. ( T GrpHom U ) /\ `' F e. ( U GrpHom T ) ) /\ ( G e. ( S GrpHom T ) /\ `' G e. ( T GrpHom S ) ) ) -> ( ( F o. G ) e. ( S GrpHom U ) /\ `' ( F o. G ) e. ( U GrpHom S ) ) )
10	1, 2, 9	syl2anb 496	. 2 |- ( ( F e. ( T GrpIso U ) /\ G e. ( S GrpIso T ) ) -> ( ( F o. G ) e. ( S GrpHom U ) /\ `' ( F o. G ) e. ( U GrpHom S ) ) )
11		isgim2 17707	. 2 |- ( ( F o. G ) e. ( S GrpIso U ) <-> ( ( F o. G ) e. ( S GrpHom U ) /\ `' ( F o. G ) e. ( U GrpHom S ) ) )
12	10, 11	sylibr 224	1 |- ( ( F e. ( T GrpIso U ) /\ G e. ( S GrpIso T ) ) -> ( F o. G ) e. ( S GrpIso U ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   `'ccnv 5113   o. ccom 5118  (class class class)co 6650   GrpHom cghm 17657   GrpIso cgim 17699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-gim 17701

This theorem is referenced by:  gictr  17717