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Theorem gimco 19299
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 19296 . . 3 (𝐹 ∈ (𝑇 GrpIso 𝑈) ↔ (𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐹 ∈ (𝑈 GrpHom 𝑇)))
2 isgim2 19296 . . 3 (𝐺 ∈ (𝑆 GrpIso 𝑇) ↔ (𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)))
3 ghmco 19267 . . . . 5 ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpHom 𝑈))
4 cnvco 5899 . . . . . 6 (𝐹𝐺) = (𝐺𝐹)
5 ghmco 19267 . . . . . . 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 2843 . . . . 5 ((𝐹 ∈ (𝑈 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)) → (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆))
83, 7anim12i 613 . . . 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 660 . . 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 598 . 2 ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → ((𝐹𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆)))
11 isgim2 19296 . 2 ((𝐹𝐺) ∈ (𝑆 GrpIso 𝑈) ↔ ((𝐹𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆)))
1210, 11sylibr 234 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 2106  ccnv 5688  ccom 5693  (class class class)co 7431   GrpHom cghm 19243   GrpIso cgim 19288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-ghm 19244  df-gim 19290
This theorem is referenced by:  gictr  19307
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