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Theorem conjnmzb 19214

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 4080	. . . 4 ⊢ 𝑁 ⊆ 𝑋
3		simpr 483	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝐴 ∈ 𝑁)
4	2, 3	sselid 3980	. . 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 19213	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 = ran 𝐹)
10	4, 9	jca 510	. 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 2815	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝐴 + 𝑤) ∈ ran 𝐹))
14		subgrcl 19093	. . . . . . . . . . . . 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 19089	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
18	17	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
19	18	sselda 3982	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
20	5, 6, 7	grpaddsubass 18993	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
21	15, 16, 19, 16, 20	syl13anc 1369	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
22	21	eqeq1d 2730	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → (((𝐴 + 𝑥) − 𝐴) = (𝐴 + 𝑤) ↔ (𝐴 + (𝑥 − 𝐴)) = (𝐴 + 𝑤)))
23	5, 7	grpsubcl 18983	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑥 − 𝐴) ∈ 𝑋)
24	15, 19, 16, 23	syl3anc 1368	. . . . . . . . . . 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 18964	. . . . . . . . . . 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 1369	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + (𝑥 − 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 − 𝐴) = 𝑤))
28	5, 6, 7	grpsubadd 18991	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑥 − 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
29	15, 19, 16, 25, 28	syl13anc 1369	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝑥 − 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
30	22, 27, 29	3bitrd 304	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → (((𝐴 + 𝑥) − 𝐴) = (𝐴 + 𝑤) ↔ (𝑤 + 𝐴) = 𝑥))
31		eqcom 2735	. . . . . . . . 9 ⊢ ((𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴) ↔ ((𝐴 + 𝑥) − 𝐴) = (𝐴 + 𝑤))
32		eqcom 2735	. . . . . . . . 9 ⊢ (𝑥 = (𝑤 + 𝐴) ↔ (𝑤 + 𝐴) = 𝑥)
33	30, 31, 32	3bitr4g 313	. . . . . . . 8 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴) ↔ 𝑥 = (𝑤 + 𝐴)))
34	33	rexbidva 3174	. . . . . . 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 7459	. . . . . . 7 ⊢ (𝐴 + 𝑤) ∈ V
37		eqeq1 2732	. . . . . . . 8 ⊢ (𝑦 = (𝐴 + 𝑤) → (𝑦 = ((𝐴 + 𝑥) − 𝐴) ↔ (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴)))
38	37	rexbidv 3176	. . . . . . 7 ⊢ (𝑦 = (𝐴 + 𝑤) → (∃𝑥 ∈ 𝑆 𝑦 = ((𝐴 + 𝑥) − 𝐴) ↔ ∃𝑥 ∈ 𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴)))
39	8	rnmpt 5961	. . . . . . 7 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = ((𝐴 + 𝑥) − 𝐴)}
40	36, 38, 39	elab2 3673	. . . . . 6 ⊢ ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴))
41		risset 3228	. . . . . 6 ⊢ ((𝑤 + 𝐴) ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑆 𝑥 = (𝑤 + 𝐴))
42	35, 40, 41	3bitr4g 313	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ (𝑤 + 𝐴) ∈ 𝑆))
43	13, 42	bitrd 278	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
44	43	ralrimiva 3143	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) → ∀𝑤 ∈ 𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
45	1	elnmz 19125	. . 3 ⊢ (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑤 ∈ 𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆)))
46	11, 44, 45	sylanbrc 581	. 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)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ∃wrex 3067 {crab 3430 ⊆ wss 3949 ↦ cmpt 5235 ran crn 5683 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +_gcplusg 17240 Grpcgrp 18897 -_gcsg 18899 SubGrpcsubg 19082

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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085

This theorem is referenced by: sylow3lem6 19594

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