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Theorem abliso 29027
 Description: The image of an Abelian group by a group isomorphism is also Abelian. (Contributed by Thierry Arnoux, 8-Mar-2018.)
Assertion
Ref Expression
abliso ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)

Proof of Theorem abliso
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gimghm 17529 . . . 4 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2 ghmgrp2 17486 . . . 4 (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
31, 2syl 17 . . 3 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝑁 ∈ Grp)
43adantl 481 . 2 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Grp)
5 grpmnd 17252 . . . 4 (𝑁 ∈ Grp → 𝑁 ∈ Mnd)
64, 5syl 17 . . 3 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Mnd)
7 simpll 786 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑀 ∈ Abel)
8 eqid 2610 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
9 eqid 2610 . . . . . . . . . . . 12 (Base‘𝑁) = (Base‘𝑁)
108, 9gimf1o 17528 . . . . . . . . . . 11 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
11 f1ocnv 6062 . . . . . . . . . . 11 (𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) → 𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀))
12 f1of 6050 . . . . . . . . . . 11 (𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀) → 𝐹:(Base‘𝑁)⟶(Base‘𝑀))
1310, 11, 123syl 18 . . . . . . . . . 10 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹:(Base‘𝑁)⟶(Base‘𝑀))
1413ad2antlr 759 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑁)⟶(Base‘𝑀))
15 simprl 790 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑥 ∈ (Base‘𝑁))
1614, 15ffvelrnd 6268 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹𝑥) ∈ (Base‘𝑀))
17 simprr 792 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑦 ∈ (Base‘𝑁))
1814, 17ffvelrnd 6268 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹𝑦) ∈ (Base‘𝑀))
19 eqid 2610 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
208, 19ablcom 18033 . . . . . . . 8 ((𝑀 ∈ Abel ∧ (𝐹𝑥) ∈ (Base‘𝑀) ∧ (𝐹𝑦) ∈ (Base‘𝑀)) → ((𝐹𝑥)(+g𝑀)(𝐹𝑦)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
217, 16, 18, 20syl3anc 1318 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ((𝐹𝑥)(+g𝑀)(𝐹𝑦)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
22 gimcnv 17532 . . . . . . . . . 10 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑁 GrpIso 𝑀))
2322ad2antlr 759 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹 ∈ (𝑁 GrpIso 𝑀))
24 gimghm 17529 . . . . . . . . 9 (𝐹 ∈ (𝑁 GrpIso 𝑀) → 𝐹 ∈ (𝑁 GrpHom 𝑀))
2523, 24syl 17 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹 ∈ (𝑁 GrpHom 𝑀))
26 eqid 2610 . . . . . . . . 9 (+g𝑁) = (+g𝑁)
279, 26, 19ghmlin 17488 . . . . . . . 8 ((𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝐹‘(𝑥(+g𝑁)𝑦)) = ((𝐹𝑥)(+g𝑀)(𝐹𝑦)))
2825, 15, 17, 27syl3anc 1318 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑥(+g𝑁)𝑦)) = ((𝐹𝑥)(+g𝑀)(𝐹𝑦)))
299, 26, 19ghmlin 17488 . . . . . . . 8 ((𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝐹‘(𝑦(+g𝑁)𝑥)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
3025, 17, 15, 29syl3anc 1318 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑦(+g𝑁)𝑥)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
3121, 28, 303eqtr4d 2654 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑥(+g𝑁)𝑦)) = (𝐹‘(𝑦(+g𝑁)𝑥)))
3231fveq2d 6107 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑥(+g𝑁)𝑦))) = (𝐹‘(𝐹‘(𝑦(+g𝑁)𝑥))))
3310ad2antlr 759 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
343ad2antlr 759 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑁 ∈ Grp)
359, 26grpcl 17253 . . . . . . 7 ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥(+g𝑁)𝑦) ∈ (Base‘𝑁))
3634, 15, 17, 35syl3anc 1318 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g𝑁)𝑦) ∈ (Base‘𝑁))
37 f1ocnvfv2 6433 . . . . . 6 ((𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) ∧ (𝑥(+g𝑁)𝑦) ∈ (Base‘𝑁)) → (𝐹‘(𝐹‘(𝑥(+g𝑁)𝑦))) = (𝑥(+g𝑁)𝑦))
3833, 36, 37syl2anc 691 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑥(+g𝑁)𝑦))) = (𝑥(+g𝑁)𝑦))
399, 26grpcl 17253 . . . . . . 7 ((𝑁 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝑦(+g𝑁)𝑥) ∈ (Base‘𝑁))
4034, 17, 15, 39syl3anc 1318 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑦(+g𝑁)𝑥) ∈ (Base‘𝑁))
41 f1ocnvfv2 6433 . . . . . 6 ((𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) ∧ (𝑦(+g𝑁)𝑥) ∈ (Base‘𝑁)) → (𝐹‘(𝐹‘(𝑦(+g𝑁)𝑥))) = (𝑦(+g𝑁)𝑥))
4233, 40, 41syl2anc 691 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑦(+g𝑁)𝑥))) = (𝑦(+g𝑁)𝑥))
4332, 38, 423eqtr3d 2652 . . . 4 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥))
4443ralrimivva 2954 . . 3 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥))
459, 26iscmn 18023 . . 3 (𝑁 ∈ CMnd ↔ (𝑁 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥)))
466, 44, 45sylanbrc 695 . 2 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ CMnd)
47 isabl 18020 . 2 (𝑁 ∈ Abel ↔ (𝑁 ∈ Grp ∧ 𝑁 ∈ CMnd))
484, 46, 47sylanbrc 695 1 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ◡ccnv 5037  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Mndcmnd 17117  Grpcgrp 17245   GrpHom cghm 17480   GrpIso cgim 17522  CMndccmn 18016  Abelcabl 18017 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-ghm 17481  df-gim 17524  df-cmn 18018  df-abl 18019 This theorem is referenced by: (None)
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