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Theorem abliso 29681
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 17700 . . . 4 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2 ghmgrp2 17657 . . . 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)
5 grpmnd 17423 . . . 4 (𝑁 ∈ Grp → 𝑁 ∈ Mnd)
64, 5syl 17 . . 3 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Mnd)
7 simpll 790 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑀 ∈ Abel)
8 eqid 2621 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
9 eqid 2621 . . . . . . . . . . . 12 (Base‘𝑁) = (Base‘𝑁)
108, 9gimf1o 17699 . . . . . . . . . . 11 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
11 f1ocnv 6147 . . . . . . . . . . 11 (𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) → 𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀))
12 f1of 6135 . . . . . . . . . . 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 763 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑁)⟶(Base‘𝑀))
15 simprl 794 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑥 ∈ (Base‘𝑁))
1614, 15ffvelrnd 6358 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹𝑥) ∈ (Base‘𝑀))
17 simprr 796 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑦 ∈ (Base‘𝑁))
1814, 17ffvelrnd 6358 . . . . . . . 8 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹𝑦) ∈ (Base‘𝑀))
19 eqid 2621 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
208, 19ablcom 18204 . . . . . . . 8 ((𝑀 ∈ Abel ∧ (𝐹𝑥) ∈ (Base‘𝑀) ∧ (𝐹𝑦) ∈ (Base‘𝑀)) → ((𝐹𝑥)(+g𝑀)(𝐹𝑦)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
217, 16, 18, 20syl3anc 1325 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ((𝐹𝑥)(+g𝑀)(𝐹𝑦)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
22 gimcnv 17703 . . . . . . . . . 10 (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑁 GrpIso 𝑀))
2322ad2antlr 763 . . . . . . . . 9 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹 ∈ (𝑁 GrpIso 𝑀))
24 gimghm 17700 . . . . . . . . 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 2621 . . . . . . . . 9 (+g𝑁) = (+g𝑁)
279, 26, 19ghmlin 17659 . . . . . . . 8 ((𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝐹‘(𝑥(+g𝑁)𝑦)) = ((𝐹𝑥)(+g𝑀)(𝐹𝑦)))
2825, 15, 17, 27syl3anc 1325 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑥(+g𝑁)𝑦)) = ((𝐹𝑥)(+g𝑀)(𝐹𝑦)))
299, 26, 19ghmlin 17659 . . . . . . . 8 ((𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝐹‘(𝑦(+g𝑁)𝑥)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
3025, 17, 15, 29syl3anc 1325 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑦(+g𝑁)𝑥)) = ((𝐹𝑦)(+g𝑀)(𝐹𝑥)))
3121, 28, 303eqtr4d 2665 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝑥(+g𝑁)𝑦)) = (𝐹‘(𝑦(+g𝑁)𝑥)))
3231fveq2d 6193 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑥(+g𝑁)𝑦))) = (𝐹‘(𝐹‘(𝑦(+g𝑁)𝑥))))
3310ad2antlr 763 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
343ad2antlr 763 . . . . . . 7 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑁 ∈ Grp)
359, 26grpcl 17424 . . . . . . 7 ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥(+g𝑁)𝑦) ∈ (Base‘𝑁))
3634, 15, 17, 35syl3anc 1325 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g𝑁)𝑦) ∈ (Base‘𝑁))
37 f1ocnvfv2 6530 . . . . . 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 693 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑥(+g𝑁)𝑦))) = (𝑥(+g𝑁)𝑦))
399, 26grpcl 17424 . . . . . . 7 ((𝑁 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝑦(+g𝑁)𝑥) ∈ (Base‘𝑁))
4034, 17, 15, 39syl3anc 1325 . . . . . 6 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑦(+g𝑁)𝑥) ∈ (Base‘𝑁))
41 f1ocnvfv2 6530 . . . . . 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 693 . . . . 5 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(𝐹‘(𝑦(+g𝑁)𝑥))) = (𝑦(+g𝑁)𝑥))
4332, 38, 423eqtr3d 2663 . . . 4 (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥))
4443ralrimivva 2970 . . 3 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥))
459, 26iscmn 18194 . . 3 (𝑁 ∈ CMnd ↔ (𝑁 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g𝑁)𝑦) = (𝑦(+g𝑁)𝑥)))
466, 44, 45sylanbrc 698 . 2 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ CMnd)
47 isabl 18191 . 2 (𝑁 ∈ Abel ↔ (𝑁 ∈ Grp ∧ 𝑁 ∈ CMnd))
484, 46, 47sylanbrc 698 1 ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  wral 2911  ccnv 5111  wf 5882  1-1-ontowf1o 5885  cfv 5886  (class class class)co 6647  Basecbs 15851  +gcplusg 15935  Mndcmnd 17288  Grpcgrp 17416   GrpHom cghm 17651   GrpIso cgim 17693  CMndccmn 18187  Abelcabl 18188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-grp 17419  df-ghm 17652  df-gim 17695  df-cmn 18189  df-abl 18190
This theorem is referenced by: (None)
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