Theorem elnmz 19109

Description: Elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypothesis

Ref	Expression
elnmz.1	⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}

Assertion

Ref	Expression
elnmz	⊢ (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))

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

Allowed substitution hints:   𝐴(𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem elnmz

Step	Hyp	Ref	Expression
1		oveq2 7422	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 + 𝑦) = (𝑥 + 𝑧))
2	1	eleq1d 2813	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥 + 𝑧) ∈ 𝑆))
3		oveq1 7421	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 + 𝑥) = (𝑧 + 𝑥))
4	3	eleq1d 2813	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑦 + 𝑥) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆))
5	2, 4	bibi12d 345	. . . 4 ⊢ (𝑦 = 𝑧 → (((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆)))
6	5	cbvralvw 3229	. . 3 ⊢ (∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆))
7		oveq1 7421	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝑧) = (𝐴 + 𝑧))
8	7	eleq1d 2813	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝐴 + 𝑧) ∈ 𝑆))
9		oveq2 7422	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑧 + 𝑥) = (𝑧 + 𝐴))
10	9	eleq1d 2813	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑧 + 𝑥) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆))
11	8, 10	bibi12d 345	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))
12	11	ralbidv 3172	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑧 ∈ 𝑋 ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))
13	6, 12	bitrid 283	. 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ∀𝑧 ∈ 𝑋 ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))
14		elnmz.1	. 2 ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
15	13, 14	elrab2 3683	1 ⊢ (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3056  {crab 3427  (class class class)co 7414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417

This theorem is referenced by:  nmzbi  19110  nmzsubg  19111  ssnmz  19112  conjnmzb  19198  sylow3lem2  19574