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Theorem abliso 31302
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 18878 . . . 4 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2 ghmgrp2 18835 . . . 4 (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
31, 2syl 17 . . 3 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝑁 ∈ Grp)
43adantl 482 . 2 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Grp)
54grpmndd 18587 . . 3 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Mnd)
6 simpll 764 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑀 ∈ Abel)
7 eqid 2738 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
8 eqid 2738 . . . . . . . . . . . 12 (Base‘𝑁) = (Base‘𝑁)
97, 8gimf1o 18877 . . . . . . . . . . 11 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
10 f1ocnv 6730 . . . . . . . . . . 11 (𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) → 𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀))
11 f1of 6718 . . . . . . . . . . 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 724 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑁)⟶(Base‘𝑀))
14 simprl 768 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑥 ∈ (Base‘𝑁))
1513, 14ffvelrnd 6964 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹𝑥) ∈ (Base‘𝑀))
16 simprr 770 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑦 ∈ (Base‘𝑁))
1713, 16ffvelrnd 6964 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹𝑦) ∈ (Base‘𝑀))
18 eqid 2738 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
197, 18ablcom 19402 . . . . . . . 8 ((𝑀 ∈ Abel ∧ (𝐹𝑥) ∈ (Base‘𝑀) ∧ (𝐹𝑦) ∈ (Base‘𝑀)) → ((𝐹𝑥)(+g𝑀)(𝐹𝑦)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
206, 15, 17, 19syl3anc 1370 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ((𝐹𝑥)(+g𝑀)(𝐹𝑦)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
21 gimcnv 18881 . . . . . . . . . 10 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑁 GrpIso 𝑀))
2221ad2antlr 724 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹 ∈ (𝑁 GrpIso 𝑀))
23 gimghm 18878 . . . . . . . . 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 2738 . . . . . . . . 9 (+g𝑁) = (+g𝑁)
268, 25, 18ghmlin 18837 . . . . . . . 8 ((𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝐹‘(𝑥(+g𝑁)𝑦)) = ((𝐹𝑥)(+g𝑀)(𝐹𝑦)))
2724, 14, 16, 26syl3anc 1370 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑥(+g𝑁)𝑦)) = ((𝐹𝑥)(+g𝑀)(𝐹𝑦)))
288, 25, 18ghmlin 18837 . . . . . . . 8 ((𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝐹‘(𝑦(+g𝑁)𝑥)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
2924, 16, 14, 28syl3anc 1370 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑦(+g𝑁)𝑥)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
3020, 27, 293eqtr4d 2788 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑥(+g𝑁)𝑦)) = (𝐹‘(𝑦(+g𝑁)𝑥)))
3130fveq2d 6780 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑥(+g𝑁)𝑦))) = (𝐹‘(𝐹‘(𝑦(+g𝑁)𝑥))))
329ad2antlr 724 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
333ad2antlr 724 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑁 ∈ Grp)
348, 25grpcl 18583 . . . . . . 7 ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥(+g𝑁)𝑦) ∈ (Base‘𝑁))
3533, 14, 16, 34syl3anc 1370 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g𝑁)𝑦) ∈ (Base‘𝑁))
36 f1ocnvfv2 7151 . . . . . 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 584 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑥(+g𝑁)𝑦))) = (𝑥(+g𝑁)𝑦))
388, 25grpcl 18583 . . . . . . 7 ((𝑁 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝑦(+g𝑁)𝑥) ∈ (Base‘𝑁))
3933, 16, 14, 38syl3anc 1370 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑦(+g𝑁)𝑥) ∈ (Base‘𝑁))
40 f1ocnvfv2 7151 . . . . . 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 584 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑦(+g𝑁)𝑥))) = (𝑦(+g𝑁)𝑥))
4231, 37, 413eqtr3d 2786 . . . 4 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥))
4342ralrimivva 3111 . . 3 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥))
448, 25iscmn 19392 . . 3 (𝑁 ∈ CMnd ↔ (𝑁 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥)))
455, 43, 44sylanbrc 583 . 2 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ CMnd)
46 isabl 19388 . 2 (𝑁 ∈ Abel ↔ (𝑁 ∈ Grp ∧ 𝑁 ∈ CMnd))
474, 45, 46sylanbrc 583 1 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wral 3064  ccnv 5590  wf 6431  1-1-ontowf1o 6434  cfv 6435  (class class class)co 7277  Basecbs 16910  +gcplusg 16960  Mndcmnd 18383  Grpcgrp 18575   GrpHom cghm 18829   GrpIso cgim 18871  CMndccmn 19384  Abelcabl 19385
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5211  ax-sep 5225  ax-nul 5232  ax-pow 5290  ax-pr 5354  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-f1 6440  df-fo 6441  df-f1o 6442  df-fv 6443  df-ov 7280  df-oprab 7281  df-mpo 7282  df-mgm 18324  df-sgrp 18373  df-mnd 18384  df-grp 18578  df-ghm 18830  df-gim 18873  df-cmn 19386  df-abl 19387
This theorem is referenced by: (None)
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