Theorem elnmz 18445

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 7190	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 + 𝑦) = (𝑥 + 𝑧))
2	1	eleq1d 2818	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥 + 𝑧) ∈ 𝑆))
3		oveq1 7189	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 + 𝑥) = (𝑧 + 𝑥))
4	3	eleq1d 2818	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑦 + 𝑥) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆))
5	2, 4	bibi12d 349	. . . 4 ⊢ (𝑦 = 𝑧 → (((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆)))
6	5	cbvralvw 3350	. . 3 ⊢ (∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆))
7		oveq1 7189	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝑧) = (𝐴 + 𝑧))
8	7	eleq1d 2818	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝐴 + 𝑧) ∈ 𝑆))
9		oveq2 7190	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑧 + 𝑥) = (𝑧 + 𝐴))
10	9	eleq1d 2818	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑧 + 𝑥) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆))
11	8, 10	bibi12d 349	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))
12	11	ralbidv 3110	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑧 ∈ 𝑋 ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))
13	6, 12	syl5bb 286	. 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 3596	1 ⊢ (𝐴 ∈ 𝑁 ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114  ∀wral 3054  {crab 3058  (class class class)co 7182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rab 3063  df-v 3402  df-un 3858  df-in 3860  df-ss 3870  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-iota 6307  df-fv 6357  df-ov 7185

This theorem is referenced by:  nmzbi  18446  nmzsubg  18447  ssnmz  18448  conjnmzb  18523  sylow3lem2  18883