Theorem conjnmzb 18395

Description: Alternative condition for elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
conjghm.x	⊢ 𝑋 = (Base‘𝐺)
conjghm.p	⊢ + = (+g‘𝐺)
conjghm.m	⊢ − = (-g‘𝐺)
conjsubg.f	⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))
conjnmz.1	⊢ 𝑁 = {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}

Assertion

Ref	Expression
conjnmzb	⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)))

Distinct variable groups:   𝑥,𝑦, −   𝑥,𝑧, + ,𝑦   𝑥,𝐴,𝑦,𝑧   𝑦,𝐹,𝑧   𝑥,𝑁   𝑥,𝐺,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Allowed substitution hints:   𝐹(𝑥)   − (𝑧)   𝑁(𝑦,𝑧)

Proof of Theorem conjnmzb

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		conjnmz.1	. . . . 5 ⊢ 𝑁 = {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}
2	1	ssrab3 4059	. . . 4 ⊢ 𝑁 ⊆ 𝑋
3		simpr 487	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝐴 ∈ 𝑁)
4	2, 3	sseldi 3967	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝐴 ∈ 𝑋)
5		conjghm.x	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
6		conjghm.p	. . . 4 ⊢ + = (+g‘𝐺)
7		conjghm.m	. . . 4 ⊢ − = (-g‘𝐺)
8		conjsubg.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))
9	5, 6, 7, 8, 1	conjnmz 18394	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 = ran 𝐹)
10	4, 9	jca 514	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹))
11		simprl 769	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) → 𝐴 ∈ 𝑋)
12		simplrr 776	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → 𝑆 = ran 𝐹)
13	12	eleq2d 2900	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝐴 + 𝑤) ∈ ran 𝐹))
14		subgrcl 18286	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
15	14	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝐺 ∈ Grp)
16		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑋)
17	5	subgss 18282	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
18	17	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
19	18	sselda 3969	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
20	5, 6, 7	grpaddsubass 18191	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
21	15, 16, 19, 16, 20	syl13anc 1368	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
22	21	eqeq1d 2825	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → (((𝐴 + 𝑥) − 𝐴) = (𝐴 + 𝑤) ↔ (𝐴 + (𝑥 − 𝐴)) = (𝐴 + 𝑤)))
23	5, 7	grpsubcl 18181	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑥 − 𝐴) ∈ 𝑋)
24	15, 19, 16, 23	syl3anc 1367	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → (𝑥 − 𝐴) ∈ 𝑋)
25		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝑤 ∈ 𝑋)
26	5, 6	grplcan 18163	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ ((𝑥 − 𝐴) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + (𝑥 − 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 − 𝐴) = 𝑤))
27	15, 24, 25, 16, 26	syl13anc 1368	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + (𝑥 − 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 − 𝐴) = 𝑤))
28	5, 6, 7	grpsubadd 18189	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑥 − 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
29	15, 19, 16, 25, 28	syl13anc 1368	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝑥 − 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
30	22, 27, 29	3bitrd 307	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → (((𝐴 + 𝑥) − 𝐴) = (𝐴 + 𝑤) ↔ (𝑤 + 𝐴) = 𝑥))
31		eqcom 2830	. . . . . . . . 9 ⊢ ((𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴) ↔ ((𝐴 + 𝑥) − 𝐴) = (𝐴 + 𝑤))
32		eqcom 2830	. . . . . . . . 9 ⊢ (𝑥 = (𝑤 + 𝐴) ↔ (𝑤 + 𝐴) = 𝑥)
33	30, 31, 32	3bitr4g 316	. . . . . . . 8 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
34	33	rexbidva 3298	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → (∃𝑥 ∈ 𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴) ↔ ∃𝑥 ∈ 𝑆 𝑥 = (𝑤 + 𝐴)))
35	34	adantlrr 719	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → (∃𝑥 ∈ 𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴) ↔ ∃𝑥 ∈ 𝑆 𝑥 = (𝑤 + 𝐴)))
36		ovex 7191	. . . . . . 7 ⊢ (𝐴 + 𝑤) ∈ V
37		eqeq1 2827	. . . . . . . 8 ⊢ (𝑦 = (𝐴 + 𝑤) → (𝑦 = ((𝐴 + 𝑥) − 𝐴) ↔ (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴)))
38	37	rexbidv 3299	. . . . . . 7 ⊢ (𝑦 = (𝐴 + 𝑤) → (∃𝑥 ∈ 𝑆 𝑦 = ((𝐴 + 𝑥) − 𝐴) ↔ ∃𝑥 ∈ 𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴)))
39	8	rnmpt 5829	. . . . . . 7 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = ((𝐴 + 𝑥) − 𝐴)}
40	36, 38, 39	elab2 3672	. . . . . 6 ⊢ ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴))
41		risset 3269	. . . . . 6 ⊢ ((𝑤 + 𝐴) ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑆 𝑥 = (𝑤 + 𝐴))
42	35, 40, 41	3bitr4g 316	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ (𝑤 + 𝐴) ∈ 𝑆))
43	13, 42	bitrd 281	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
44	43	ralrimiva 3184	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) → ∀𝑤 ∈ 𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
45	1	elnmz 18317	. . 3 ⊢ (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑤 ∈ 𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆)))
46	11, 44, 45	sylanbrc 585	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) → 𝐴 ∈ 𝑁)
47	10, 46	impbida 799	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  {crab 3144   ⊆ wss 3938   ↦ cmpt 5148  ran crn 5558  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  +_gcplusg 16567  Grpcgrp 18105  -_gcsg 18107  SubGrpcsubg 18275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-minusg 18109  df-sbg 18110  df-subg 18278

This theorem is referenced by:  sylow3lem6  18759