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Theorem ghmf1o 17898
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 17871 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
2 ghmgrp1 17870 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
31, 2jca 501 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
43adantr 466 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
5 f1ocnv 6290 . . . . . 6 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
65adantl 467 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌1-1-onto𝑋)
7 f1of 6278 . . . . 5 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
86, 7syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌𝑋)
9 simpll 750 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
108adantr 466 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝐹:𝑌𝑋)
11 simprl 754 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝑥𝑌)
1210, 11ffvelrnd 6503 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹𝑥) ∈ 𝑋)
13 simprr 756 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝑦𝑌)
1410, 13ffvelrnd 6503 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹𝑦) ∈ 𝑋)
15 ghmf1o.x . . . . . . . . 9 𝑋 = (Base‘𝑆)
16 eqid 2771 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
17 eqid 2771 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
1815, 16, 17ghmlin 17873 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → (𝐹‘((𝐹𝑥)(+g𝑆)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))))
199, 12, 14, 18syl3anc 1476 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘((𝐹𝑥)(+g𝑆)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))))
20 simplr 752 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝐹:𝑋1-1-onto𝑌)
21 f1ocnvfv2 6676 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
2220, 11, 21syl2anc 573 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘(𝐹𝑥)) = 𝑥)
23 f1ocnvfv2 6676 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑦𝑌) → (𝐹‘(𝐹𝑦)) = 𝑦)
2420, 13, 23syl2anc 573 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2522, 24oveq12d 6811 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑇)𝑦))
2619, 25eqtrd 2805 . . . . . 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 17638 . . . . . . . 8 ((𝑆 ∈ Grp ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∈ 𝑋)
2927, 12, 14, 28syl3anc 1476 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∈ 𝑋)
30 f1ocnvfv 6677 . . . . . . 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 573 . . . . . 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 3120 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → ∀𝑥𝑌𝑦𝑌 (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
348, 33jca 501 . . 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 17868 . . 3 (𝐹 ∈ (𝑇 GrpHom 𝑆) ↔ ((𝑇 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝑌𝑋 ∧ ∀𝑥𝑌𝑦𝑌 (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
374, 34, 36sylanbrc 572 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 GrpHom 𝑆))
3815, 35ghmf 17872 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋𝑌)
3938adantr 466 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋𝑌)
40 ffn 6185 . . . 4 (𝐹:𝑋𝑌𝐹 Fn 𝑋)
4139, 40syl 17 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑋)
4235, 15ghmf 17872 . . . . 5 (𝐹 ∈ (𝑇 GrpHom 𝑆) → 𝐹:𝑌𝑋)
4342adantl 467 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑌𝑋)
44 ffn 6185 . . . 4 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
4543, 44syl 17 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑌)
46 dff1o4 6286 . . 3 (𝐹:𝑋1-1-onto𝑌 ↔ (𝐹 Fn 𝑋𝐹 Fn 𝑌))
4741, 45, 46sylanbrc 572 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋1-1-onto𝑌)
4837, 47impbida 802 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 382   = wceq 1631  wcel 2145  wral 3061  ccnv 5248   Fn wfn 6026  wf 6027  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  Grpcgrp 17630   GrpHom cghm 17865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-ghm 17866
This theorem is referenced by:  isgim2  17915  rhmf1o  18942  lmhmf1o  19259  rnghmf1o  42431
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