Theorem abliso 32697

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.

Step	Hyp	Ref	Expression
1		gimghm 19186	. . . 4 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2		ghmgrp2 19141	. . . 4 ⊢ (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
3	1, 2	syl 17	. . 3 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝑁 ∈ Grp)
4	3	adantl 481	. 2 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Grp)
5	4	grpmndd 18873	. . 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 2726	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
8		eqid 2726	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
9	7, 8	gimf1o 19185	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
10		f1ocnv 6838	. . . . . . . . . . 11 ⊢ (𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) → ◡𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀))
11		f1of 6826	. . . . . . . . . . 11 ⊢ (◡𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀) → ◡𝐹:(Base‘𝑁)⟶(Base‘𝑀))
12	9, 10, 11	3syl 18	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → ◡𝐹:(Base‘𝑁)⟶(Base‘𝑀))
13	12	ad2antlr 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))
15	13, 14	ffvelcdmd 7080	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘𝑥) ∈ (Base‘𝑀))
16		simprr 770	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑦 ∈ (Base‘𝑁))
17	13, 16	ffvelcdmd 7080	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘𝑦) ∈ (Base‘𝑀))
18		eqid 2726	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
19	7, 18	ablcom 19716	. . . . . . . 8 ⊢ ((𝑀 ∈ Abel ∧ (◡𝐹‘𝑥) ∈ (Base‘𝑀) ∧ (◡𝐹‘𝑦) ∈ (Base‘𝑀)) → ((◡𝐹‘𝑥)(+g‘𝑀)(◡𝐹‘𝑦)) = ((◡𝐹‘𝑦)(+g‘𝑀)(◡𝐹‘𝑥)))
20	6, 15, 17, 19	syl3anc 1368	. . . . . . 7 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ((◡𝐹‘𝑥)(+g‘𝑀)(◡𝐹‘𝑦)) = ((◡𝐹‘𝑦)(+g‘𝑀)(◡𝐹‘𝑥)))
21		gimcnv 19189	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → ◡𝐹 ∈ (𝑁 GrpIso 𝑀))
22	21	ad2antlr 724	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ◡𝐹 ∈ (𝑁 GrpIso 𝑀))
23		gimghm 19186	. . . . . . . . 9 ⊢ (◡𝐹 ∈ (𝑁 GrpIso 𝑀) → ◡𝐹 ∈ (𝑁 GrpHom 𝑀))
24	22, 23	syl 17	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ◡𝐹 ∈ (𝑁 GrpHom 𝑀))
25		eqid 2726	. . . . . . . . 9 ⊢ (+g‘𝑁) = (+g‘𝑁)
26	8, 25, 18	ghmlin 19143	. . . . . . . 8 ⊢ ((◡𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (◡𝐹‘(𝑥(+g‘𝑁)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑀)(◡𝐹‘𝑦)))
27	24, 14, 16, 26	syl3anc 1368	. . . . . . 7 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘(𝑥(+g‘𝑁)𝑦)) = ((◡𝐹‘𝑥)(+g‘𝑀)(◡𝐹‘𝑦)))
28	8, 25, 18	ghmlin 19143	. . . . . . . 8 ⊢ ((◡𝐹 ∈ (𝑁 GrpHom 𝑀) ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (◡𝐹‘(𝑦(+g‘𝑁)𝑥)) = ((◡𝐹‘𝑦)(+g‘𝑀)(◡𝐹‘𝑥)))
29	24, 16, 14, 28	syl3anc 1368	. . . . . . 7 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘(𝑦(+g‘𝑁)𝑥)) = ((◡𝐹‘𝑦)(+g‘𝑀)(◡𝐹‘𝑥)))
30	20, 27, 29	3eqtr4d 2776	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘(𝑥(+g‘𝑁)𝑦)) = (◡𝐹‘(𝑦(+g‘𝑁)𝑥)))
31	30	fveq2d 6888	. . . . 5 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(◡𝐹‘(𝑥(+g‘𝑁)𝑦))) = (𝐹‘(◡𝐹‘(𝑦(+g‘𝑁)𝑥))))
32	9	ad2antlr 724	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
33	3	ad2antlr 724	. . . . . . 7 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑁 ∈ Grp)
34	8, 25	grpcl 18868	. . . . . . 7 ⊢ ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥(+g‘𝑁)𝑦) ∈ (Base‘𝑁))
35	33, 14, 16, 34	syl3anc 1368	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g‘𝑁)𝑦) ∈ (Base‘𝑁))
36		f1ocnvfv2 7270	. . . . . 6 ⊢ ((𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) ∧ (𝑥(+g‘𝑁)𝑦) ∈ (Base‘𝑁)) → (𝐹‘(◡𝐹‘(𝑥(+g‘𝑁)𝑦))) = (𝑥(+g‘𝑁)𝑦))
37	32, 35, 36	syl2anc 583	. . . . 5 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(◡𝐹‘(𝑥(+g‘𝑁)𝑦))) = (𝑥(+g‘𝑁)𝑦))
38	8, 25	grpcl 18868	. . . . . . 7 ⊢ ((𝑁 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝑦(+g‘𝑁)𝑥) ∈ (Base‘𝑁))
39	33, 16, 14, 38	syl3anc 1368	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑦(+g‘𝑁)𝑥) ∈ (Base‘𝑁))
40		f1ocnvfv2 7270	. . . . . 6 ⊢ ((𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) ∧ (𝑦(+g‘𝑁)𝑥) ∈ (Base‘𝑁)) → (𝐹‘(◡𝐹‘(𝑦(+g‘𝑁)𝑥))) = (𝑦(+g‘𝑁)𝑥))
41	32, 39, 40	syl2anc 583	. . . . 5 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝐹‘(◡𝐹‘(𝑦(+g‘𝑁)𝑥))) = (𝑦(+g‘𝑁)𝑥))
42	31, 37, 41	3eqtr3d 2774	. . . 4 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g‘𝑁)𝑦) = (𝑦(+g‘𝑁)𝑥))
43	42	ralrimivva 3194	. . 3 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g‘𝑁)𝑦) = (𝑦(+g‘𝑁)𝑥))
44	8, 25	iscmn 19706	. . 3 ⊢ (𝑁 ∈ CMnd ↔ (𝑁 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g‘𝑁)𝑦) = (𝑦(+g‘𝑁)𝑥)))
45	5, 43, 44	sylanbrc 582	. 2 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ CMnd)
46		isabl 19701	. 2 ⊢ (𝑁 ∈ Abel ↔ (𝑁 ∈ Grp ∧ 𝑁 ∈ CMnd))
47	4, 45, 46	sylanbrc 582	1 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ^◡ccnv 5668  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7404  Basecbs 17150  +_gcplusg 17203  Mndcmnd 18664  Grpcgrp 18860   GrpHom cghm 19135   GrpIso cgim 19179  CMndccmn 19697  Abelcabl 19698

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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-grp 18863  df-ghm 19136  df-gim 19181  df-cmn 19699  df-abl 19700

This theorem is referenced by: (None)