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Theorem ghmf1o 18469
 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 18442 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
2 ghmgrp1 18441 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
31, 2jca 515 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
43adantr 484 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
5 f1ocnv 6619 . . . . . 6 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
65adantl 485 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌1-1-onto𝑋)
7 f1of 6607 . . . . 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 484 . . . . . . . . 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, 11ffvelrnd 6849 . . . . . . . 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, 13ffvelrnd 6849 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹𝑦) ∈ 𝑋)
15 ghmf1o.x . . . . . . . . 9 𝑋 = (Base‘𝑆)
16 eqid 2758 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
17 eqid 2758 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
1815, 16, 17ghmlin 18444 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → (𝐹‘((𝐹𝑥)(+g𝑆)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))))
199, 12, 14, 18syl3anc 1368 . . . . . . 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 7032 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
2220, 11, 21syl2anc 587 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘(𝐹𝑥)) = 𝑥)
23 f1ocnvfv2 7032 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑦𝑌) → (𝐹‘(𝐹𝑦)) = 𝑦)
2420, 13, 23syl2anc 587 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2522, 24oveq12d 7174 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))) = (𝑥(+g𝑇)𝑦))
2619, 25eqtrd 2793 . . . . . 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 18191 . . . . . . . 8 ((𝑆 ∈ Grp ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∈ 𝑋)
2927, 12, 14, 28syl3anc 1368 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∈ 𝑋)
30 f1ocnvfv 7033 . . . . . . 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 587 . . . . . 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 515 . . 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 18439 . . 3 (𝐹 ∈ (𝑇 GrpHom 𝑆) ↔ ((𝑇 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝑌𝑋 ∧ ∀𝑥𝑌𝑦𝑌 (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
374, 34, 36sylanbrc 586 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 GrpHom 𝑆))
3815, 35ghmf 18443 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋𝑌)
3938adantr 484 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋𝑌)
4039ffnd 6504 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑋)
4135, 15ghmf 18443 . . . . 5 (𝐹 ∈ (𝑇 GrpHom 𝑆) → 𝐹:𝑌𝑋)
4241adantl 485 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑌𝑋)
4342ffnd 6504 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑌)
44 dff1o4 6615 . . 3 (𝐹:𝑋1-1-onto𝑌 ↔ (𝐹 Fn 𝑋𝐹 Fn 𝑌))
4540, 43, 44sylanbrc 586 . 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ◡ccnv 5527   Fn wfn 6335  ⟶wf 6336  –1-1-onto→wf1o 6339  ‘cfv 6340  (class class class)co 7156  Basecbs 16555  +gcplusg 16637  Grpcgrp 18183   GrpHom cghm 18436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-mgm 17932  df-sgrp 17981  df-mnd 17992  df-grp 18186  df-ghm 18437 This theorem is referenced by:  isgim2  18486  rhmf1o  19569  lmhmf1o  19900  rnghmf1o  44953
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