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Theorem abliso 30740
 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 18400 . . . 4 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2 ghmgrp2 18357 . . . 4 (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
31, 2syl 17 . . 3 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝑁 ∈ Grp)
43adantl 485 . 2 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Grp)
5 grpmnd 18105 . . . 4 (𝑁 ∈ Grp → 𝑁 ∈ Mnd)
64, 5syl 17 . . 3 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Mnd)
7 simpll 766 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑀 ∈ Abel)
8 eqid 2798 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
9 eqid 2798 . . . . . . . . . . . 12 (Base‘𝑁) = (Base‘𝑁)
108, 9gimf1o 18399 . . . . . . . . . . 11 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
11 f1ocnv 6603 . . . . . . . . . . 11 (𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) → 𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀))
12 f1of 6591 . . . . . . . . . . 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 726 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑁)⟶(Base‘𝑀))
15 simprl 770 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑥 ∈ (Base‘𝑁))
1614, 15ffvelrnd 6830 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹𝑥) ∈ (Base‘𝑀))
17 simprr 772 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑦 ∈ (Base‘𝑁))
1814, 17ffvelrnd 6830 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹𝑦) ∈ (Base‘𝑀))
19 eqid 2798 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
208, 19ablcom 18920 . . . . . . . 8 ((𝑀 ∈ Abel ∧ (𝐹𝑥) ∈ (Base‘𝑀) ∧ (𝐹𝑦) ∈ (Base‘𝑀)) → ((𝐹𝑥)(+g𝑀)(𝐹𝑦)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
217, 16, 18, 20syl3anc 1368 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ((𝐹𝑥)(+g𝑀)(𝐹𝑦)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
22 gimcnv 18403 . . . . . . . . . 10 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑁 GrpIso 𝑀))
2322ad2antlr 726 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹 ∈ (𝑁 GrpIso 𝑀))
24 gimghm 18400 . . . . . . . . 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 2798 . . . . . . . . 9 (+g𝑁) = (+g𝑁)
279, 26, 19ghmlin 18359 . . . . . . . 8 ((𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝐹‘(𝑥(+g𝑁)𝑦)) = ((𝐹𝑥)(+g𝑀)(𝐹𝑦)))
2825, 15, 17, 27syl3anc 1368 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑥(+g𝑁)𝑦)) = ((𝐹𝑥)(+g𝑀)(𝐹𝑦)))
299, 26, 19ghmlin 18359 . . . . . . . 8 ((𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝐹‘(𝑦(+g𝑁)𝑥)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
3025, 17, 15, 29syl3anc 1368 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑦(+g𝑁)𝑥)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
3121, 28, 303eqtr4d 2843 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑥(+g𝑁)𝑦)) = (𝐹‘(𝑦(+g𝑁)𝑥)))
3231fveq2d 6650 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑥(+g𝑁)𝑦))) = (𝐹‘(𝐹‘(𝑦(+g𝑁)𝑥))))
3310ad2antlr 726 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
343ad2antlr 726 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑁 ∈ Grp)
359, 26grpcl 18106 . . . . . . 7 ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥(+g𝑁)𝑦) ∈ (Base‘𝑁))
3634, 15, 17, 35syl3anc 1368 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g𝑁)𝑦) ∈ (Base‘𝑁))
37 f1ocnvfv2 7013 . . . . . 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 587 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑥(+g𝑁)𝑦))) = (𝑥(+g𝑁)𝑦))
399, 26grpcl 18106 . . . . . . 7 ((𝑁 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝑦(+g𝑁)𝑥) ∈ (Base‘𝑁))
4034, 17, 15, 39syl3anc 1368 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑦(+g𝑁)𝑥) ∈ (Base‘𝑁))
41 f1ocnvfv2 7013 . . . . . 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 587 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑦(+g𝑁)𝑥))) = (𝑦(+g𝑁)𝑥))
4332, 38, 423eqtr3d 2841 . . . 4 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥))
4443ralrimivva 3156 . . 3 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥))
459, 26iscmn 18910 . . 3 (𝑁 ∈ CMnd ↔ (𝑁 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥)))
466, 44, 45sylanbrc 586 . 2 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ CMnd)
47 isabl 18906 . 2 (𝑁 ∈ Abel ↔ (𝑁 ∈ Grp ∧ 𝑁 ∈ CMnd))
484, 46, 47sylanbrc 586 1 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ◡ccnv 5519  ⟶wf 6321  –1-1-onto→wf1o 6324  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  +gcplusg 16560  Mndcmnd 17906  Grpcgrp 18098   GrpHom cghm 18351   GrpIso cgim 18393  CMndccmn 18902  Abelcabl 18903 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 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18101  df-ghm 18352  df-gim 18395  df-cmn 18904  df-abl 18905 This theorem is referenced by: (None)
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