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Theorem gimco 19017
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 19014 . . 3 (𝐹 ∈ (𝑇 GrpIso 𝑈) ↔ (𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐹 ∈ (𝑈 GrpHom 𝑇)))
2 isgim2 19014 . . 3 (𝐺 ∈ (𝑆 GrpIso 𝑇) ↔ (𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)))
3 ghmco 18987 . . . . 5 ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpHom 𝑈))
4 cnvco 5840 . . . . . 6 (𝐹𝐺) = (𝐺𝐹)
5 ghmco 18987 . . . . . . 7 ((𝐺 ∈ (𝑇 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑈 GrpHom 𝑇)) → (𝐺𝐹) ∈ (𝑈 GrpHom 𝑆))
65ancoms 460 . . . . . 6 ((𝐹 ∈ (𝑈 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)) → (𝐺𝐹) ∈ (𝑈 GrpHom 𝑆))
74, 6eqeltrid 2843 . . . . 5 ((𝐹 ∈ (𝑈 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)) → (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆))
83, 7anim12i 614 . . . 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 659 . . 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 599 . 2 ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → ((𝐹𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆)))
11 isgim2 19014 . 2 ((𝐹𝐺) ∈ (𝑆 GrpIso 𝑈) ↔ ((𝐹𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆)))
1210, 11sylibr 233 1 ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpIso 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  ccnv 5631  ccom 5636  (class class class)co 7352   GrpHom cghm 18964   GrpIso cgim 19006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7308  df-ov 7355  df-oprab 7356  df-mpo 7357  df-map 8726  df-0g 17283  df-mgm 18457  df-sgrp 18506  df-mnd 18517  df-mhm 18561  df-grp 18711  df-ghm 18965  df-gim 19008
This theorem is referenced by:  gictr  19024
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