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Theorem nsgbi 18959
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 18957 . . . 4 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
43simprbi 497 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))
5 oveq1 7364 . . . . . 6 (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
65eleq1d 2822 . . . . 5 (𝑥 = 𝐴 → ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝐴 + 𝑦) ∈ 𝑆))
7 oveq2 7365 . . . . . 6 (𝑥 = 𝐴 → (𝑦 + 𝑥) = (𝑦 + 𝐴))
87eleq1d 2822 . . . . 5 (𝑥 = 𝐴 → ((𝑦 + 𝑥) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆))
96, 8bibi12d 345 . . . 4 (𝑥 = 𝐴 → (((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆)))
10 oveq2 7365 . . . . . 6 (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵))
1110eleq1d 2822 . . . . 5 (𝑦 = 𝐵 → ((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝐴 + 𝐵) ∈ 𝑆))
12 oveq1 7364 . . . . . 6 (𝑦 = 𝐵 → (𝑦 + 𝐴) = (𝐵 + 𝐴))
1312eleq1d 2822 . . . . 5 (𝑦 = 𝐵 → ((𝑦 + 𝐴) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))
1411, 13bibi12d 345 . . . 4 (𝑦 = 𝐵 → (((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆) ↔ ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)))
159, 14rspc2v 3590 . . 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 1116 1 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  SubGrpcsubg 18922  NrmSGrpcnsg 18923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-subg 18925  df-nsg 18926
This theorem is referenced by:  nsgconj  18961  eqgcpbl  18984
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