Theorem conjnmzb 19293

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 4105	. . . 4 ⊢ 𝑁 ⊆ 𝑋
3		simpr 484	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝐴 ∈ 𝑁)
4	2, 3	sselid 4006	. . 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 19292	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 = ran 𝐹)
10	4, 9	jca 511	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹))
11		simprl 770	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) → 𝐴 ∈ 𝑋)
12		simplrr 777	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → 𝑆 = ran 𝐹)
13	12	eleq2d 2830	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝐴 + 𝑤) ∈ ran 𝐹))
14		subgrcl 19171	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
15	14	ad3antrrr 729	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝐺 ∈ Grp)
16		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑋)
17	5	subgss 19167	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
18	17	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
19	18	sselda 4008	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
20	5, 6, 7	grpaddsubass 19070	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
21	15, 16, 19, 16, 20	syl13anc 1372	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
22	21	eqeq1d 2742	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → (((𝐴 + 𝑥) − 𝐴) = (𝐴 + 𝑤) ↔ (𝐴 + (𝑥 − 𝐴)) = (𝐴 + 𝑤)))
23	5, 7	grpsubcl 19060	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑥 − 𝐴) ∈ 𝑋)
24	15, 19, 16, 23	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → (𝑥 − 𝐴) ∈ 𝑋)
25		simplr 768	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝑤 ∈ 𝑋)
26	5, 6	grplcan 19040	. . . . . . . . . . 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 1372	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + (𝑥 − 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 − 𝐴) = 𝑤))
28	5, 6, 7	grpsubadd 19068	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑥 − 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
29	15, 19, 16, 25, 28	syl13anc 1372	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝑥 − 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
30	22, 27, 29	3bitrd 305	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → (((𝐴 + 𝑥) − 𝐴) = (𝐴 + 𝑤) ↔ (𝑤 + 𝐴) = 𝑥))
31		eqcom 2747	. . . . . . . . 9 ⊢ ((𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴) ↔ ((𝐴 + 𝑥) − 𝐴) = (𝐴 + 𝑤))
32		eqcom 2747	. . . . . . . . 9 ⊢ (𝑥 = (𝑤 + 𝐴) ↔ (𝑤 + 𝐴) = 𝑥)
33	30, 31, 32	3bitr4g 314	. . . . . . . 8 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
34	33	rexbidva 3183	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → (∃𝑥 ∈ 𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴) ↔ ∃𝑥 ∈ 𝑆 𝑥 = (𝑤 + 𝐴)))
35	34	adantlrr 720	. . . . . 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 7481	. . . . . . 7 ⊢ (𝐴 + 𝑤) ∈ V
37		eqeq1 2744	. . . . . . . 8 ⊢ (𝑦 = (𝐴 + 𝑤) → (𝑦 = ((𝐴 + 𝑥) − 𝐴) ↔ (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴)))
38	37	rexbidv 3185	. . . . . . 7 ⊢ (𝑦 = (𝐴 + 𝑤) → (∃𝑥 ∈ 𝑆 𝑦 = ((𝐴 + 𝑥) − 𝐴) ↔ ∃𝑥 ∈ 𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴)))
39	8	rnmpt 5980	. . . . . . 7 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = ((𝐴 + 𝑥) − 𝐴)}
40	36, 38, 39	elab2 3698	. . . . . 6 ⊢ ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴))
41		risset 3239	. . . . . 6 ⊢ ((𝑤 + 𝐴) ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑆 𝑥 = (𝑤 + 𝐴))
42	35, 40, 41	3bitr4g 314	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ (𝑤 + 𝐴) ∈ 𝑆))
43	13, 42	bitrd 279	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
44	43	ralrimiva 3152	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) → ∀𝑤 ∈ 𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
45	1	elnmz 19203	. . 3 ⊢ (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑤 ∈ 𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆)))
46	11, 44, 45	sylanbrc 582	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) → 𝐴 ∈ 𝑁)
47	10, 46	impbida 800	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443   ⊆ wss 3976   ↦ cmpt 5249  ran crn 5701  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  Grpcgrp 18973  -_gcsg 18975  SubGrpcsubg 19160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-sbg 18978  df-subg 19163

This theorem is referenced by:  sylow3lem6  19674