Theorem abliso 33215

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 19305	. . . 4 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2		ghmgrp2 19260	. . . 4 ⊢ (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
3	1, 2	syl 17	. . 3 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝑁 ∈ Grp)
4	3	adantl 485	. 2 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Grp)
5	4	grpmndd 18989	. . 3 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Mnd)
6		simpll 776	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑀 ∈ Abel)
7		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
8		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
9	7, 8	gimf1o 19304	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
10		f1ocnv 6820	. . . . . . . . . . 11 ⊢ (𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁) → ◡𝐹:(Base‘𝑁)–1-1-onto→(Base‘𝑀))
11		f1of 6807	. . . . . . . . . . 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 737	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ◡𝐹:(Base‘𝑁)⟶(Base‘𝑀))
14		simprl 780	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑥 ∈ (Base‘𝑁))
15	13, 14	ffvelcdmd 7067	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘𝑥) ∈ (Base‘𝑀))
16		simprr 782	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑦 ∈ (Base‘𝑁))
17	13, 16	ffvelcdmd 7067	. . . . . . . 8 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (◡𝐹‘𝑦) ∈ (Base‘𝑀))
18		eqid 2763	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
19	7, 18	ablcom 19840	. . . . . . . 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 1391	. . . . . . 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 19308	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑀 GrpIso 𝑁) → ◡𝐹 ∈ (𝑁 GrpIso 𝑀))
22	21	ad2antlr 737	. . . . . . . . 9 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → ◡𝐹 ∈ (𝑁 GrpIso 𝑀))
23		gimghm 19305	. . . . . . . . 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 2763	. . . . . . . . 9 ⊢ (+g‘𝑁) = (+g‘𝑁)
26	8, 25, 18	ghmlin 19262	. . . . . . . 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 1391	. . . . . . 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 19262	. . . . . . . 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 1391	. . . . . . 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 2808	. . . . . 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 6872	. . . . 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 737	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝐹:(Base‘𝑀)–1-1-onto→(Base‘𝑁))
33	3	ad2antlr 737	. . . . . . 7 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → 𝑁 ∈ Grp)
34	8, 25	grpcl 18984	. . . . . . 7 ⊢ ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥(+g‘𝑁)𝑦) ∈ (Base‘𝑁))
35	33, 14, 16, 34	syl3anc 1391	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g‘𝑁)𝑦) ∈ (Base‘𝑁))
36		f1ocnvfv2 7262	. . . . . 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 593	. . . . 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 18984	. . . . . . 7 ⊢ ((𝑁 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑁) ∧ 𝑥 ∈ (Base‘𝑁)) → (𝑦(+g‘𝑁)𝑥) ∈ (Base‘𝑁))
39	33, 16, 14, 38	syl3anc 1391	. . . . . 6 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑦(+g‘𝑁)𝑥) ∈ (Base‘𝑁))
40		f1ocnvfv2 7262	. . . . . 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 593	. . . . 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 2806	. . . 4 ⊢ (((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥(+g‘𝑁)𝑦) = (𝑦(+g‘𝑁)𝑥))
43	42	ralrimivva 3206	. . 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 19830	. . 3 ⊢ (𝑁 ∈ CMnd ↔ (𝑁 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑁)∀𝑦 ∈ (Base‘𝑁)(𝑥(+g‘𝑁)𝑦) = (𝑦(+g‘𝑁)𝑥)))
45	5, 43, 44	sylanbrc 592	. 2 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ CMnd)
46		isabl 19825	. 2 ⊢ (𝑁 ∈ Abel ↔ (𝑁 ∈ Grp ∧ 𝑁 ∈ CMnd))
47	4, 45, 46	sylanbrc 592	1 ⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  ∀wral 3077  ^◡ccnv 5647  ⟶wf 6518  –1-1-onto→wf1o 6521  ‘cfv 6522  (class class class)co 7397  Basecbs 17246  +_gcplusg 17287  Mndcmnd 18769  Grpcgrp 18976   GrpHom cghm 19254   GrpIso cgim 19298  CMndccmn 19821  Abelcabl 19822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-1st 7971  df-2nd 7972  df-map 8811  df-mgm 18675  df-sgrp 18754  df-mnd 18770  df-grp 18979  df-ghm 19255  df-gim 19300  df-cmn 19823  df-abl 19824

This theorem is referenced by: (None)