Theorem nsgbi 18301

Description: Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
isnsg.1	⊢ 𝑋 = (Base‘𝐺)
isnsg.2	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
nsgbi	⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))

Proof of Theorem nsgbi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isnsg.1	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
2		isnsg.2	. . . . 5 ⊢ + = (+g‘𝐺)
3	1, 2	isnsg 18299	. . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
4	3	simprbi 500	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))
5		oveq1 7142	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
6	5	eleq1d 2874	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝐴 + 𝑦) ∈ 𝑆))
7		oveq2 7143	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑦 + 𝑥) = (𝑦 + 𝐴))
8	7	eleq1d 2874	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 + 𝑥) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆))
9	6, 8	bibi12d 349	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆)))
10		oveq2 7143	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵))
11	10	eleq1d 2874	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝐴 + 𝐵) ∈ 𝑆))
12		oveq1 7142	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 + 𝐴) = (𝐵 + 𝐴))
13	12	eleq1d 2874	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑦 + 𝐴) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))
14	11, 13	bibi12d 349	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆) ↔ ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)))
15	9, 14	rspc2v 3581	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)))
16	4, 15	syl5com 31	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)))
17	16	3impib 1113	1 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  SubGrpcsubg 18265  NrmSGrpcnsg 18266

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-subg 18268  df-nsg 18269

This theorem is referenced by:  nsgconj  18303  eqgcpbl  18326