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Theorem isnsg 19173
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 19142 . . 3 NrmSGrp = (𝑔 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑔) ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
21mptrcl 7025 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp)
3 subgrcl 19149 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
43adantr 480 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)) → 𝐺 ∈ Grp)
5 fveq2 6906 . . . . . 6 (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
6 fvexd 6921 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) ∈ V)
7 fveq2 6906 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
8 isnsg.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
97, 8eqtr4di 2795 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
10 fvexd 6921 . . . . . . . 8 ((𝑔 = 𝐺𝑏 = 𝑋) → (+g𝑔) ∈ V)
11 simpl 482 . . . . . . . . . 10 ((𝑔 = 𝐺𝑏 = 𝑋) → 𝑔 = 𝐺)
1211fveq2d 6910 . . . . . . . . 9 ((𝑔 = 𝐺𝑏 = 𝑋) → (+g𝑔) = (+g𝐺))
13 isnsg.2 . . . . . . . . 9 + = (+g𝐺)
1412, 13eqtr4di 2795 . . . . . . . 8 ((𝑔 = 𝐺𝑏 = 𝑋) → (+g𝑔) = + )
15 simplr 769 . . . . . . . . 9 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑏 = 𝑋)
16 simpr 484 . . . . . . . . . . . . 13 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑝 = + )
1716oveqd 7448 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
1817eleq1d 2826 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑠))
1916oveqd 7448 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑦𝑝𝑥) = (𝑦 + 𝑥))
2019eleq1d 2826 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑦𝑝𝑥) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠))
2118, 20bibi12d 345 . . . . . . . . . 10 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
2215, 21raleqbidv 3346 . . . . . . . . 9 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
2315, 22raleqbidv 3346 . . . . . . . 8 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
2410, 14, 23sbcied2 3833 . . . . . . 7 ((𝑔 = 𝐺𝑏 = 𝑋) → ([(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
256, 9, 24sbcied2 3833 . . . . . 6 (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
265, 25rabeqbidv 3455 . . . . 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)})
27 fvex 6919 . . . . . 6 (SubGrp‘𝐺) ∈ V
2827rabex 5339 . . . . 5 {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ∈ V
2926, 1, 28fvmpt 7016 . . . 4 (𝐺 ∈ Grp → (NrmSGrp‘𝐺) = {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)})
3029eleq2d 2827 . . 3 (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}))
31 eleq2 2830 . . . . . 6 (𝑠 = 𝑆 → ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑆))
32 eleq2 2830 . . . . . 6 (𝑠 = 𝑆 → ((𝑦 + 𝑥) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑆))
3331, 32bibi12d 345 . . . . 5 (𝑠 = 𝑆 → (((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠) ↔ ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
34332ralbidv 3221 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
3534elrab 3692 . . 3 (𝑆 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
3630, 35bitrdi 287 . 2 (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))))
372, 4, 36pm5.21nii 378 1 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  [wsbc 3788  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951  SubGrpcsubg 19138  NrmSGrpcnsg 19139
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-subg 19141  df-nsg 19142
This theorem is referenced by:  isnsg2  19174  nsgbi  19175  nsgsubg  19176  isnsg4  19185  nmznsg  19186  ablnsg  19865  rzgrp  21641
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