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Theorem isnsg 17830
Description: Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
isnsg.1 𝑋 = (Base‘𝐺)
isnsg.2 + = (+g𝐺)
Assertion
Ref Expression
isnsg (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥, + ,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Proof of Theorem isnsg
Dummy variables 𝑔 𝑏 𝑝 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nsg 17799 . . . 4 NrmSGrp = (𝑔 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑔) ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
21dmmptss 5775 . . 3 dom NrmSGrp ⊆ Grp
3 elfvdm 6361 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ dom NrmSGrp)
42, 3sseldi 3748 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp)
5 subgrcl 17806 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
65adantr 466 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)) → 𝐺 ∈ Grp)
7 fveq2 6332 . . . . . 6 (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
8 fvexd 6344 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) ∈ V)
9 fveq2 6332 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
10 isnsg.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
119, 10syl6eqr 2822 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
12 fvexd 6344 . . . . . . . 8 ((𝑔 = 𝐺𝑏 = 𝑋) → (+g𝑔) ∈ V)
13 simpl 468 . . . . . . . . . 10 ((𝑔 = 𝐺𝑏 = 𝑋) → 𝑔 = 𝐺)
1413fveq2d 6336 . . . . . . . . 9 ((𝑔 = 𝐺𝑏 = 𝑋) → (+g𝑔) = (+g𝐺))
15 isnsg.2 . . . . . . . . 9 + = (+g𝐺)
1614, 15syl6eqr 2822 . . . . . . . 8 ((𝑔 = 𝐺𝑏 = 𝑋) → (+g𝑔) = + )
17 simplr 744 . . . . . . . . 9 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑏 = 𝑋)
18 simpr 471 . . . . . . . . . . . . 13 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑝 = + )
1918oveqd 6809 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
2019eleq1d 2834 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑠))
2118oveqd 6809 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑦𝑝𝑥) = (𝑦 + 𝑥))
2221eleq1d 2834 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑦𝑝𝑥) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠))
2320, 22bibi12d 334 . . . . . . . . . 10 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
2417, 23raleqbidv 3300 . . . . . . . . 9 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
2517, 24raleqbidv 3300 . . . . . . . 8 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
2612, 16, 25sbcied2 3623 . . . . . . 7 ((𝑔 = 𝐺𝑏 = 𝑋) → ([(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
278, 11, 26sbcied2 3623 . . . . . 6 (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
287, 27rabeqbidv 3344 . . . . 5 (𝑔 = 𝐺 → {𝑠 ∈ (SubGrp‘𝑔) ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)} = {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)})
29 fvex 6342 . . . . . 6 (SubGrp‘𝐺) ∈ V
3029rabex 4943 . . . . 5 {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ∈ V
3128, 1, 30fvmpt 6424 . . . 4 (𝐺 ∈ Grp → (NrmSGrp‘𝐺) = {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)})
3231eleq2d 2835 . . 3 (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}))
33 eleq2 2838 . . . . . 6 (𝑠 = 𝑆 → ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑆))
34 eleq2 2838 . . . . . 6 (𝑠 = 𝑆 → ((𝑦 + 𝑥) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑆))
3533, 34bibi12d 334 . . . . 5 (𝑠 = 𝑆 → (((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠) ↔ ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
36352ralbidv 3137 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
3736elrab 3513 . . 3 (𝑆 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
3832, 37syl6bb 276 . 2 (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))))
394, 6, 38pm5.21nii 367 1 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1630  wcel 2144  wral 3060  {crab 3064  Vcvv 3349  [wsbc 3585  dom cdm 5249  cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  Grpcgrp 17629  SubGrpcsubg 17795  NrmSGrpcnsg 17796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-subg 17798  df-nsg 17799
This theorem is referenced by:  isnsg2  17831  nsgbi  17832  nsgsubg  17833  isnsg4  17844  nmznsg  17845  ablnsg  18456  rzgrp  24520
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