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Theorem nsgbi 18288
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 18286 . . . 4 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
43simprbi 500 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))
5 oveq1 7137 . . . . . 6 (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
65eleq1d 2896 . . . . 5 (𝑥 = 𝐴 → ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝐴 + 𝑦) ∈ 𝑆))
7 oveq2 7138 . . . . . 6 (𝑥 = 𝐴 → (𝑦 + 𝑥) = (𝑦 + 𝐴))
87eleq1d 2896 . . . . 5 (𝑥 = 𝐴 → ((𝑦 + 𝑥) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆))
96, 8bibi12d 349 . . . 4 (𝑥 = 𝐴 → (((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆)))
10 oveq2 7138 . . . . . 6 (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵))
1110eleq1d 2896 . . . . 5 (𝑦 = 𝐵 → ((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝐴 + 𝐵) ∈ 𝑆))
12 oveq1 7137 . . . . . 6 (𝑦 = 𝐵 → (𝑦 + 𝐴) = (𝐵 + 𝐴))
1312eleq1d 2896 . . . . 5 (𝑦 = 𝐵 → ((𝑦 + 𝐴) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))
1411, 13bibi12d 349 . . . 4 (𝑦 = 𝐵 → (((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆) ↔ ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)))
159, 14rspc2v 3610 . . 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 1113 1 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  cfv 6328  (class class class)co 7130  Basecbs 16462  +gcplusg 16544  SubGrpcsubg 18252  NrmSGrpcnsg 18253
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-subg 18255  df-nsg 18256
This theorem is referenced by:  nsgconj  18290  eqgcpbl  18313
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