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Theorem abliso 32777
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 19225 . . . 4 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2 ghmgrp2 19180 . . . 4 (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
31, 2syl 17 . . 3 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝑁 ∈ Grp)
43adantl 480 . 2 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Grp)
54grpmndd 18910 . . 3 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Mnd)
6 simpll 765 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑀 ∈ Abel)
7 eqid 2728 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
8 eqid 2728 . . . . . . . . . . . 12 (Base‘𝑁) = (Base‘𝑁)
97, 8gimf1o 19224 . . . . . . . . . . 11 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
10 f1ocnv 6856 . . . . . . . . . . 11 (𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) → 𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀))
11 f1of 6844 . . . . . . . . . . 11 (𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀) → 𝐹:(Base‘𝑁)⟶(Base‘𝑀))
129, 10, 113syl 18 . . . . . . . . . 10 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹:(Base‘𝑁)⟶(Base‘𝑀))
1312ad2antlr 725 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑁)⟶(Base‘𝑀))
14 simprl 769 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑥 ∈ (Base‘𝑁))
1513, 14ffvelcdmd 7100 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹𝑥) ∈ (Base‘𝑀))
16 simprr 771 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑦 ∈ (Base‘𝑁))
1713, 16ffvelcdmd 7100 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹𝑦) ∈ (Base‘𝑀))
18 eqid 2728 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
197, 18ablcom 19761 . . . . . . . 8 ((𝑀 ∈ Abel ∧ (𝐹𝑥) ∈ (Base‘𝑀) ∧ (𝐹𝑦) ∈ (Base‘𝑀)) → ((𝐹𝑥)(+g𝑀)(𝐹𝑦)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
206, 15, 17, 19syl3anc 1368 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ((𝐹𝑥)(+g𝑀)(𝐹𝑦)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
21 gimcnv 19228 . . . . . . . . . 10 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑁 GrpIso 𝑀))
2221ad2antlr 725 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹 ∈ (𝑁 GrpIso 𝑀))
23 gimghm 19225 . . . . . . . . 9 (𝐹 ∈ (𝑁 GrpIso 𝑀) → 𝐹 ∈ (𝑁 GrpHom 𝑀))
2422, 23syl 17 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹 ∈ (𝑁 GrpHom 𝑀))
25 eqid 2728 . . . . . . . . 9 (+g𝑁) = (+g𝑁)
268, 25, 18ghmlin 19182 . . . . . . . 8 ((𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝐹‘(𝑥(+g𝑁)𝑦)) = ((𝐹𝑥)(+g𝑀)(𝐹𝑦)))
2724, 14, 16, 26syl3anc 1368 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑥(+g𝑁)𝑦)) = ((𝐹𝑥)(+g𝑀)(𝐹𝑦)))
288, 25, 18ghmlin 19182 . . . . . . . 8 ((𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝐹‘(𝑦(+g𝑁)𝑥)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
2924, 16, 14, 28syl3anc 1368 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑦(+g𝑁)𝑥)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
3020, 27, 293eqtr4d 2778 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑥(+g𝑁)𝑦)) = (𝐹‘(𝑦(+g𝑁)𝑥)))
3130fveq2d 6906 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑥(+g𝑁)𝑦))) = (𝐹‘(𝐹‘(𝑦(+g𝑁)𝑥))))
329ad2antlr 725 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
333ad2antlr 725 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑁 ∈ Grp)
348, 25grpcl 18905 . . . . . . 7 ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥(+g𝑁)𝑦) ∈ (Base‘𝑁))
3533, 14, 16, 34syl3anc 1368 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g𝑁)𝑦) ∈ (Base‘𝑁))
36 f1ocnvfv2 7292 . . . . . 6 ((𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) ∧ (𝑥(+g𝑁)𝑦) ∈ (Base‘𝑁)) → (𝐹‘(𝐹‘(𝑥(+g𝑁)𝑦))) = (𝑥(+g𝑁)𝑦))
3732, 35, 36syl2anc 582 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑥(+g𝑁)𝑦))) = (𝑥(+g𝑁)𝑦))
388, 25grpcl 18905 . . . . . . 7 ((𝑁 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝑦(+g𝑁)𝑥) ∈ (Base‘𝑁))
3933, 16, 14, 38syl3anc 1368 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑦(+g𝑁)𝑥) ∈ (Base‘𝑁))
40 f1ocnvfv2 7292 . . . . . 6 ((𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) ∧ (𝑦(+g𝑁)𝑥) ∈ (Base‘𝑁)) → (𝐹‘(𝐹‘(𝑦(+g𝑁)𝑥))) = (𝑦(+g𝑁)𝑥))
4132, 39, 40syl2anc 582 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑦(+g𝑁)𝑥))) = (𝑦(+g𝑁)𝑥))
4231, 37, 413eqtr3d 2776 . . . 4 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥))
4342ralrimivva 3198 . . 3 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥))
448, 25iscmn 19751 . . 3 (𝑁 ∈ CMnd ↔ (𝑁 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥)))
455, 43, 44sylanbrc 581 . 2 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ CMnd)
46 isabl 19746 . 2 (𝑁 ∈ Abel ↔ (𝑁 ∈ Grp ∧ 𝑁 ∈ CMnd))
474, 45, 46sylanbrc 581 1 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3058  ccnv 5681  wf 6549  1-1-ontowf1o 6552  cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  Mndcmnd 18701  Grpcgrp 18897   GrpHom cghm 19174   GrpIso cgim 19218  CMndccmn 19742  Abelcabl 19743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-ghm 19175  df-gim 19220  df-cmn 19744  df-abl 19745
This theorem is referenced by: (None)
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