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Theorem ghmf1o 17611
Description: A bijective group homomorphism is an isomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
ghmf1o.x 𝑋 = (Base‘𝑆)
ghmf1o.y 𝑌 = (Base‘𝑇)
Assertion
Ref Expression
ghmf1o (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 GrpHom 𝑆)))

Proof of Theorem ghmf1o
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmgrp2 17584 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
2 ghmgrp1 17583 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
31, 2jca 554 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
43adantr 481 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
5 f1ocnv 6106 . . . . . 6 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
65adantl 482 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌1-1-onto𝑋)
7 f1of 6094 . . . . 5 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
86, 7syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌𝑋)
9 simpll 789 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
108adantr 481 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝐹:𝑌𝑋)
11 simprl 793 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝑥𝑌)
1210, 11ffvelrnd 6316 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹𝑥) ∈ 𝑋)
13 simprr 795 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝑦𝑌)
1410, 13ffvelrnd 6316 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹𝑦) ∈ 𝑋)
15 ghmf1o.x . . . . . . . . 9 𝑋 = (Base‘𝑆)
16 eqid 2621 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
17 eqid 2621 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
1815, 16, 17ghmlin 17586 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → (𝐹‘((𝐹𝑥)(+g𝑆)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))))
199, 12, 14, 18syl3anc 1323 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘((𝐹𝑥)(+g𝑆)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))))
20 simplr 791 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝐹:𝑋1-1-onto𝑌)
21 f1ocnvfv2 6487 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
2220, 11, 21syl2anc 692 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘(𝐹𝑥)) = 𝑥)
23 f1ocnvfv2 6487 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑦𝑌) → (𝐹‘(𝐹𝑦)) = 𝑦)
2420, 13, 23syl2anc 692 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2522, 24oveq12d 6622 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑇)𝑦))
2619, 25eqtrd 2655 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘((𝐹𝑥)(+g𝑆)(𝐹𝑦))) = (𝑥(+g𝑇)𝑦))
279, 2syl 17 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝑆 ∈ Grp)
2815, 16grpcl 17351 . . . . . . . 8 ((𝑆 ∈ Grp ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∈ 𝑋)
2927, 12, 14, 28syl3anc 1323 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∈ 𝑋)
30 f1ocnvfv 6488 . . . . . . 7 ((𝐹:𝑋1-1-onto𝑌 ∧ ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∈ 𝑋) → ((𝐹‘((𝐹𝑥)(+g𝑆)(𝐹𝑦))) = (𝑥(+g𝑇)𝑦) → (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))))
3120, 29, 30syl2anc 692 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐹‘((𝐹𝑥)(+g𝑆)(𝐹𝑦))) = (𝑥(+g𝑇)𝑦) → (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))))
3226, 31mpd 15 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
3332ralrimivva 2965 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → ∀𝑥𝑌𝑦𝑌 (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
348, 33jca 554 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (𝐹:𝑌𝑋 ∧ ∀𝑥𝑌𝑦𝑌 (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦))))
35 ghmf1o.y . . . 4 𝑌 = (Base‘𝑇)
3635, 15, 17, 16isghm 17581 . . 3 (𝐹 ∈ (𝑇 GrpHom 𝑆) ↔ ((𝑇 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝑌𝑋 ∧ ∀𝑥𝑌𝑦𝑌 (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
374, 34, 36sylanbrc 697 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 GrpHom 𝑆))
3815, 35ghmf 17585 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋𝑌)
3938adantr 481 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋𝑌)
40 ffn 6002 . . . 4 (𝐹:𝑋𝑌𝐹 Fn 𝑋)
4139, 40syl 17 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑋)
4235, 15ghmf 17585 . . . . 5 (𝐹 ∈ (𝑇 GrpHom 𝑆) → 𝐹:𝑌𝑋)
4342adantl 482 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑌𝑋)
44 ffn 6002 . . . 4 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
4543, 44syl 17 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑌)
46 dff1o4 6102 . . 3 (𝐹:𝑋1-1-onto𝑌 ↔ (𝐹 Fn 𝑋𝐹 Fn 𝑌))
4741, 45, 46sylanbrc 697 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋1-1-onto𝑌)
4837, 47impbida 876 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 GrpHom 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  ccnv 5073   Fn wfn 5842  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Grpcgrp 17343   GrpHom cghm 17578
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-ghm 17579
This theorem is referenced by:  isgim2  17628  rhmf1o  18653  lmhmf1o  18965  rnghmf1o  41188
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