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Theorem gimco 18338
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 18335 . . 3 (𝐹 ∈ (𝑇 GrpIso 𝑈) ↔ (𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐹 ∈ (𝑈 GrpHom 𝑇)))
2 isgim2 18335 . . 3 (𝐺 ∈ (𝑆 GrpIso 𝑇) ↔ (𝐺 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)))
3 ghmco 18308 . . . . 5 ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpHom 𝑈))
4 cnvco 5755 . . . . . 6 (𝐹𝐺) = (𝐺𝐹)
5 ghmco 18308 . . . . . . 7 ((𝐺 ∈ (𝑇 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑈 GrpHom 𝑇)) → (𝐺𝐹) ∈ (𝑈 GrpHom 𝑆))
65ancoms 459 . . . . . 6 ((𝐹 ∈ (𝑈 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑇 GrpHom 𝑆)) → (𝐺𝐹) ∈ (𝑈 GrpHom 𝑆))
74, 6eqeltrid 2922 . . . . 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 656 . . 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 18335 . 2 ((𝐹𝐺) ∈ (𝑆 GrpIso 𝑈) ↔ ((𝐹𝐺) ∈ (𝑆 GrpHom 𝑈) ∧ (𝐹𝐺) ∈ (𝑈 GrpHom 𝑆)))
1210, 11sylibr 235 1 ((𝐹 ∈ (𝑇 GrpIso 𝑈) ∧ 𝐺 ∈ (𝑆 GrpIso 𝑇)) → (𝐹𝐺) ∈ (𝑆 GrpIso 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2107  ccnv 5553  ccom 5558  (class class class)co 7148   GrpHom cghm 18285   GrpIso cgim 18327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8398  df-0g 16705  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-mhm 17944  df-grp 18036  df-ghm 18286  df-gim 18329
This theorem is referenced by:  gictr  18345
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