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Mirrors > Home > MPE Home > Th. List > nsgbi | Structured version Visualization version GIF version |
Description: Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
Ref | Expression |
---|---|
isnsg.1 | ⊢ 𝑋 = (Base‘𝐺) |
isnsg.2 | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
nsgbi | ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnsg.1 | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
2 | isnsg.2 | . . . . 5 ⊢ + = (+g‘𝐺) | |
3 | 1, 2 | isnsg 18310 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))) |
4 | 3 | simprbi 499 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)) |
5 | oveq1 7166 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦)) | |
6 | 5 | eleq1d 2900 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝐴 + 𝑦) ∈ 𝑆)) |
7 | oveq2 7167 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑦 + 𝑥) = (𝑦 + 𝐴)) | |
8 | 7 | eleq1d 2900 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 + 𝑥) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆)) |
9 | 6, 8 | bibi12d 348 | . . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆) ↔ ((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆))) |
10 | oveq2 7167 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵)) | |
11 | 10 | eleq1d 2900 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝐴 + 𝐵) ∈ 𝑆)) |
12 | oveq1 7166 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 + 𝐴) = (𝐵 + 𝐴)) | |
13 | 12 | eleq1d 2900 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑦 + 𝐴) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)) |
14 | 11, 13 | bibi12d 348 | . . . 4 ⊢ (𝑦 = 𝐵 → (((𝐴 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝐴) ∈ 𝑆) ↔ ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆))) |
15 | 9, 14 | rspc2v 3636 | . . 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 1112 | 1 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 + 𝐵) ∈ 𝑆 ↔ (𝐵 + 𝐴) ∈ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 +gcplusg 16568 SubGrpcsubg 18276 NrmSGrpcnsg 18277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-subg 18279 df-nsg 18280 |
This theorem is referenced by: nsgconj 18314 eqgcpbl 18337 |
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