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Theorem nsgbi 17448
 Description: Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
isnsg.1 𝑋 = (Base‘𝐺)
isnsg.2 + = (+g𝐺)
Assertion
Ref Expression
nsgbi ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))

Proof of Theorem nsgbi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnsg.1 . . . . 5 𝑋 = (Base‘𝐺)
2 isnsg.2 . . . . 5 + = (+g𝐺)
31, 2isnsg 17446 . . . 4 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
43simprbi 479 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))
5 oveq1 6556 . . . . . 6 (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
65eleq1d 2672 . . . . 5 (𝑥 = 𝐴 → ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝐴 + 𝑦) ∈ 𝑆))
7 oveq2 6557 . . . . . 6 (𝑥 = 𝐴 → (𝑦 + 𝑥) = (𝑦 + 𝐴))
87eleq1d 2672 . . . . 5 (𝑥 = 𝐴 → ((𝑦 + 𝑥) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆))
96, 8bibi12d 334 . . . 4 (𝑥 = 𝐴 → (((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆)))
10 oveq2 6557 . . . . . 6 (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵))
1110eleq1d 2672 . . . . 5 (𝑦 = 𝐵 → ((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝐴 + 𝐵) ∈ 𝑆))
12 oveq1 6556 . . . . . 6 (𝑦 = 𝐵 → (𝑦 + 𝐴) = (𝐵 + 𝐴))
1312eleq1d 2672 . . . . 5 (𝑦 = 𝐵 → ((𝑦 + 𝐴) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))
1411, 13bibi12d 334 . . . 4 (𝑦 = 𝐵 → (((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆) ↔ ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)))
159, 14rspc2v 3293 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)))
164, 15syl5com 31 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) → ((𝐴𝑋𝐵𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)))
17163impib 1254 1 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  SubGrpcsubg 17411  NrmSGrpcnsg 17412 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-subg 17414  df-nsg 17415 This theorem is referenced by:  nsgconj  17450  eqgcpbl  17471
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