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Theorem gimco 17650
Description: The composition of group isomorphisms is a group isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
Assertion
Ref Expression
gimco ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpIso 𝑈))

Proof of Theorem gimco
StepHypRef Expression
1 isgim2 17647 . . 3 (𝐹 ∈ (𝑇 GrpIso 𝑈) ↔ (𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐹 ∈ (𝑈 GrpHom 𝑇)))
2 isgim2 17647 . . 3 (𝐺 ∈ (𝑆 GrpIso 𝑇) ↔ (𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)))
3 ghmco 17620 . . . . 5 ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpHom 𝑈))
4 cnvco 5278 . . . . . 6 (𝐹𝐺) = (𝐺𝐹)
5 ghmco 17620 . . . . . . 7 ((𝐺 ∈ (𝑇 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑈 GrpHom 𝑇)) → (𝐺𝐹) ∈ (𝑈 GrpHom 𝑆))
65ancoms 469 . . . . . 6 ((𝐹 ∈ (𝑈 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)) → (𝐺𝐹) ∈ (𝑈 GrpHom 𝑆))
74, 6syl5eqel 2702 . . . . 5 ((𝐹 ∈ (𝑈 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)) → (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆))
83, 7anim12i 589 . . . 4 (((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝐹 ∈ (𝑈 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆))) → ((𝐹𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆)))
98an4s 868 . . 3 (((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐹 ∈ (𝑈 GrpHom 𝑇)) ∧ (𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆))) → ((𝐹𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆)))
101, 2, 9syl2anb 496 . 2 ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → ((𝐹𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆)))
11 isgim2 17647 . 2 ((𝐹𝐺) ∈ (𝑆 GrpIso 𝑈) ↔ ((𝐹𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆)))
1210, 11sylibr 224 1 ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpIso 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  ccnv 5083  ccom 5088  (class class class)co 6615   GrpHom cghm 17597   GrpIso cgim 17639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275  df-grp 17365  df-ghm 17598  df-gim 17641
This theorem is referenced by:  gictr  17657
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