Theorem conjnmzb 18869

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 4015	. . . 4 ⊢ 𝑁 ⊆ 𝑋
3		simpr 485	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝐴 ∈ 𝑁)
4	2, 3	sselid 3919	. . 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 18868	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → 𝑆 = ran 𝐹)
10	4, 9	jca 512	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑁) → (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹))
11		simprl 768	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) → 𝐴 ∈ 𝑋)
12		simplrr 775	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → 𝑆 = ran 𝐹)
13	12	eleq2d 2824	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝐴 + 𝑤) ∈ ran 𝐹))
14		subgrcl 18760	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
15	14	ad3antrrr 727	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝐺 ∈ Grp)
16		simpllr 773	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ 𝑋)
17	5	subgss 18756	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
18	17	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
19	18	sselda 3921	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
20	5, 6, 7	grpaddsubass 18665	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
21	15, 16, 19, 16, 20	syl13anc 1371	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + 𝑥) − 𝐴) = (𝐴 + (𝑥 − 𝐴)))
22	21	eqeq1d 2740	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → (((𝐴 + 𝑥) − 𝐴) = (𝐴 + 𝑤) ↔ (𝐴 + (𝑥 − 𝐴)) = (𝐴 + 𝑤)))
23	5, 7	grpsubcl 18655	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑥 − 𝐴) ∈ 𝑋)
24	15, 19, 16, 23	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → (𝑥 − 𝐴) ∈ 𝑋)
25		simplr 766	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → 𝑤 ∈ 𝑋)
26	5, 6	grplcan 18637	. . . . . . . . . . 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 1371	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ 𝑥 ∈ 𝑆) → ((𝐴 + (𝑥 − 𝐴)) = (𝐴 + 𝑤) ↔ (𝑥 − 𝐴) = 𝑤))
28	5, 6, 7	grpsubadd 18663	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑥 − 𝐴) = 𝑤 ↔ (𝑤 + 𝐴) = 𝑥))
29	15, 19, 16, 25, 28	syl13anc 1371	. . . . . . . . . 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 2745	. . . . . . . . 9 ⊢ ((𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴) ↔ ((𝐴 + 𝑥) − 𝐴) = (𝐴 + 𝑤))
32		eqcom 2745	. . . . . . . . 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 3225	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → (∃𝑥 ∈ 𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴) ↔ ∃𝑥 ∈ 𝑆 𝑥 = (𝑤 + 𝐴)))
35	34	adantlrr 718	. . . . . 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 7308	. . . . . . 7 ⊢ (𝐴 + 𝑤) ∈ V
37		eqeq1 2742	. . . . . . . 8 ⊢ (𝑦 = (𝐴 + 𝑤) → (𝑦 = ((𝐴 + 𝑥) − 𝐴) ↔ (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴)))
38	37	rexbidv 3226	. . . . . . 7 ⊢ (𝑦 = (𝐴 + 𝑤) → (∃𝑥 ∈ 𝑆 𝑦 = ((𝐴 + 𝑥) − 𝐴) ↔ ∃𝑥 ∈ 𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴)))
39	8	rnmpt 5864	. . . . . . 7 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = ((𝐴 + 𝑥) − 𝐴)}
40	36, 38, 39	elab2 3613	. . . . . 6 ⊢ ((𝐴 + 𝑤) ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑆 (𝐴 + 𝑤) = ((𝐴 + 𝑥) − 𝐴))
41		risset 3194	. . . . . 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 278	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) ∧ 𝑤 ∈ 𝑋) → ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
44	43	ralrimiva 3103	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) → ∀𝑤 ∈ 𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆))
45	1	elnmz 18791	. . 3 ⊢ (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑤 ∈ 𝑋 ((𝐴 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝐴) ∈ 𝑆)))
46	11, 44, 45	sylanbrc 583	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)) → 𝐴 ∈ 𝑁)
47	10, 46	impbida 798	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ 𝑆 = ran 𝐹)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068   ⊆ wss 3887   ↦ cmpt 5157  ran crn 5590  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  Grpcgrp 18577  -_gcsg 18579  SubGrpcsubg 18749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752

This theorem is referenced by:  sylow3lem6  19237