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Theorem ghmf1o 19266
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 19237 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
2 ghmgrp1 19236 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
31, 2jca 511 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
43adantr 480 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → (𝑇 ∈ Grp ∧ 𝑆 ∈ Grp))
5 f1ocnv 6860 . . . . . 6 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
65adantl 481 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌1-1-onto𝑋)
7 f1of 6848 . . . . 5 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
86, 7syl 17 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹:𝑌𝑋)
9 simpll 767 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
108adantr 480 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝐹:𝑌𝑋)
11 simprl 771 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝑥𝑌)
1210, 11ffvelcdmd 7105 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹𝑥) ∈ 𝑋)
13 simprr 773 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝑦𝑌)
1410, 13ffvelcdmd 7105 . . . . . . . 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 19239 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → (𝐹‘((𝐹𝑥)(+g𝑆)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))))
199, 12, 14, 18syl3anc 1373 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘((𝐹𝑥)(+g𝑆)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(+g𝑇)(𝐹‘(𝐹𝑦))))
20 simplr 769 . . . . . . . . 9 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → 𝐹:𝑋1-1-onto𝑌)
21 f1ocnvfv2 7297 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
2220, 11, 21syl2anc 584 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘(𝐹𝑥)) = 𝑥)
23 f1ocnvfv2 7297 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑦𝑌) → (𝐹‘(𝐹𝑦)) = 𝑦)
2420, 13, 23syl2anc 584 . . . . . . . 8 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2522, 24oveq12d 7449 . . . . . . 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 18959 . . . . . . . 8 ((𝑆 ∈ Grp ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∈ 𝑋)
2927, 12, 14, 28syl3anc 1373 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) ∧ (𝑥𝑌𝑦𝑌)) → ((𝐹𝑥)(+g𝑆)(𝐹𝑦)) ∈ 𝑋)
30 f1ocnvfv 7298 . . . . . . 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 584 . . . . . 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 3202 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → ∀𝑥𝑌𝑦𝑌 (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))
348, 33jca 511 . . 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 19233 . . 3 (𝐹 ∈ (𝑇 GrpHom 𝑆) ↔ ((𝑇 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝑌𝑋 ∧ ∀𝑥𝑌𝑦𝑌 (𝐹‘(𝑥(+g𝑇)𝑦)) = ((𝐹𝑥)(+g𝑆)(𝐹𝑦)))))
374, 34, 36sylanbrc 583 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1-onto𝑌) → 𝐹 ∈ (𝑇 GrpHom 𝑆))
3815, 35ghmf 19238 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋𝑌)
3938adantr 480 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋𝑌)
4039ffnd 6737 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑋)
4135, 15ghmf 19238 . . . . 5 (𝐹 ∈ (𝑇 GrpHom 𝑆) → 𝐹:𝑌𝑋)
4241adantl 481 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑌𝑋)
4342ffnd 6737 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹 Fn 𝑌)
44 dff1o4 6856 . . 3 (𝐹:𝑋1-1-onto𝑌 ↔ (𝐹 Fn 𝑋𝐹 Fn 𝑌))
4540, 43, 44sylanbrc 583 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ (𝑇 GrpHom 𝑆)) → 𝐹:𝑋1-1-onto𝑌)
4637, 45impbida 801 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 GrpHom 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  ccnv 5684   Fn wfn 6556  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951   GrpHom cghm 19230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-ghm 19231
This theorem is referenced by:  isgim2  19283  rnghmf1o  20452  rhmf1o  20491  lmhmf1o  21045
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