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Theorem gimco 19221
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 19218 . . 3 (𝐹 ∈ (𝑇 GrpIso 𝑈) ↔ (𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐹 ∈ (𝑈 GrpHom 𝑇)))
2 isgim2 19218 . . 3 (𝐺 ∈ (𝑆 GrpIso 𝑇) ↔ (𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)))
3 ghmco 19189 . . . . 5 ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpHom 𝑈))
4 cnvco 5888 . . . . . 6 (𝐹𝐺) = (𝐺𝐹)
5 ghmco 19189 . . . . . . 7 ((𝐺 ∈ (𝑇 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑈 GrpHom 𝑇)) → (𝐺𝐹) ∈ (𝑈 GrpHom 𝑆))
65ancoms 458 . . . . . 6 ((𝐹 ∈ (𝑈 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)) → (𝐺𝐹) ∈ (𝑈 GrpHom 𝑆))
74, 6eqeltrid 2833 . . . . 5 ((𝐹 ∈ (𝑈 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)) → (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆))
83, 7anim12i 612 . . . 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 597 . 2 ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → ((𝐹𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆)))
11 isgim2 19218 . 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 395  wcel 2099  ccnv 5677  ccom 5682  (class class class)co 7420   GrpHom cghm 19166   GrpIso cgim 19210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8846  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18739  df-grp 18892  df-ghm 19167  df-gim 19212
This theorem is referenced by:  gictr  19229
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