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Theorem elnmz 17554
Description: Elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypothesis
Ref Expression
elnmz.1 𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
Assertion
Ref Expression
elnmz (𝐴𝑁 ↔ (𝐴𝑋 ∧ ∀𝑧𝑋 ((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
StepHypRef Expression
1 oveq2 6612 . . . . . 6 (𝑦 = 𝑧 → (𝑥 + 𝑦) = (𝑥 + 𝑧))
21eleq1d 2683 . . . . 5 (𝑦 = 𝑧 → ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥 + 𝑧) ∈ 𝑆))
3 oveq1 6611 . . . . . 6 (𝑦 = 𝑧 → (𝑦 + 𝑥) = (𝑧 + 𝑥))
43eleq1d 2683 . . . . 5 (𝑦 = 𝑧 → ((𝑦 + 𝑥) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆))
52, 4bibi12d 335 . . . 4 (𝑦 = 𝑧 → (((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆)))
65cbvralv 3159 . . 3 (∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ∀𝑧𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆))
7 oveq1 6611 . . . . . 6 (𝑥 = 𝐴 → (𝑥 + 𝑧) = (𝐴 + 𝑧))
87eleq1d 2683 . . . . 5 (𝑥 = 𝐴 → ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝐴 + 𝑧) ∈ 𝑆))
9 oveq2 6612 . . . . . 6 (𝑥 = 𝐴 → (𝑧 + 𝑥) = (𝑧 + 𝐴))
109eleq1d 2683 . . . . 5 (𝑥 = 𝐴 → ((𝑧 + 𝑥) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆))
118, 10bibi12d 335 . . . 4 (𝑥 = 𝐴 → (((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))
1211ralbidv 2980 . . 3 (𝑥 = 𝐴 → (∀𝑧𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑧𝑋 ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))
136, 12syl5bb 272 . 2 (𝑥 = 𝐴 → (∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ∀𝑧𝑋 ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))
14 elnmz.1 . 2 𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
1513, 14elrab2 3348 1 (𝐴𝑁 ↔ (𝐴𝑋 ∧ ∀𝑧𝑋 ((𝐴 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝐴) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  (class class class)co 6604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607
This theorem is referenced by:  nmzbi  17555  nmzsubg  17556  ssnmz  17557  conjnmzb  17616  sylow3lem2  17964
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