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Theorem ghmf1o 19039
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 19012 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
2 ghmgrp1 19011 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
31, 2jca 513 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
43adantr 482 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
5 f1ocnv 6797 . . . . . 6 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
65adantl 483 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌1-1-onto𝑋)
7 f1of 6785 . . . . 5 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
86, 7syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌𝑋)
9 simpll 766 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
108adantr 482 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝐹:𝑌𝑋)
11 simprl 770 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝑥𝑌)
1210, 11ffvelcdmd 7037 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹𝑥) ∈ 𝑋)
13 simprr 772 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝑦𝑌)
1410, 13ffvelcdmd 7037 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹𝑦) ∈ 𝑋)
15 ghmf1o.x . . . . . . . . 9 𝑋 = (Base‘𝑆)
16 eqid 2737 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
17 eqid 2737 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
1815, 16, 17ghmlin 19014 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → (𝐹‘((𝐹𝑥)(+g𝑆)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))))
199, 12, 14, 18syl3anc 1372 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘((𝐹𝑥)(+g𝑆)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))))
20 simplr 768 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝐹:𝑋1-1-onto𝑌)
21 f1ocnvfv2 7224 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
2220, 11, 21syl2anc 585 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘(𝐹𝑥)) = 𝑥)
23 f1ocnvfv2 7224 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑦𝑌) → (𝐹‘(𝐹𝑦)) = 𝑦)
2420, 13, 23syl2anc 585 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2522, 24oveq12d 7376 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑇)𝑦))
2619, 25eqtrd 2777 . . . . . 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 18757 . . . . . . . 8 ((𝑆 ∈ Grp ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∈ 𝑋)
2927, 12, 14, 28syl3anc 1372 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∈ 𝑋)
30 f1ocnvfv 7225 . . . . . . 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 585 . . . . . 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 3198 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → ∀𝑥𝑌𝑦𝑌 (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
348, 33jca 513 . . 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 19009 . . 3 (𝐹 ∈ (𝑇 GrpHom 𝑆) ↔ ((𝑇 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝑌𝑋 ∧ ∀𝑥𝑌𝑦𝑌 (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
374, 34, 36sylanbrc 584 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 GrpHom 𝑆))
3815, 35ghmf 19013 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋𝑌)
3938adantr 482 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋𝑌)
4039ffnd 6670 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑋)
4135, 15ghmf 19013 . . . . 5 (𝐹 ∈ (𝑇 GrpHom 𝑆) → 𝐹:𝑌𝑋)
4241adantl 483 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑌𝑋)
4342ffnd 6670 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑌)
44 dff1o4 6793 . . 3 (𝐹:𝑋1-1-onto𝑌 ↔ (𝐹 Fn 𝑋𝐹 Fn 𝑌))
4540, 43, 44sylanbrc 584 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋1-1-onto𝑌)
4637, 45impbida 800 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 GrpHom 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  ccnv 5633   Fn wfn 6492  wf 6493  1-1-ontowf1o 6496  cfv 6497  (class class class)co 7358  Basecbs 17084  +gcplusg 17134  Grpcgrp 18749   GrpHom cghm 19006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-mgm 18498  df-sgrp 18547  df-mnd 18558  df-grp 18752  df-ghm 19007
This theorem is referenced by:  isgim2  19056  rhmf1o  20165  lmhmf1o  20510  rnghmf1o  46208
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