MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnsg Structured version   Visualization version   GIF version

Theorem isnsg 19110
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 19079 . . 3 NrmSGrp = (𝑔 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑔) ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
21mptrcl 7014 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp)
3 subgrcl 19086 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
43adantr 480 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)) → 𝐺 ∈ Grp)
5 fveq2 6897 . . . . . 6 (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
6 fvexd 6912 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) ∈ V)
7 fveq2 6897 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
8 isnsg.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
97, 8eqtr4di 2786 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
10 fvexd 6912 . . . . . . . 8 ((𝑔 = 𝐺𝑏 = 𝑋) → (+g𝑔) ∈ V)
11 simpl 482 . . . . . . . . . 10 ((𝑔 = 𝐺𝑏 = 𝑋) → 𝑔 = 𝐺)
1211fveq2d 6901 . . . . . . . . 9 ((𝑔 = 𝐺𝑏 = 𝑋) → (+g𝑔) = (+g𝐺))
13 isnsg.2 . . . . . . . . 9 + = (+g𝐺)
1412, 13eqtr4di 2786 . . . . . . . 8 ((𝑔 = 𝐺𝑏 = 𝑋) → (+g𝑔) = + )
15 simplr 768 . . . . . . . . 9 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑏 = 𝑋)
16 simpr 484 . . . . . . . . . . . . 13 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑝 = + )
1716oveqd 7437 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
1817eleq1d 2814 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑠))
1916oveqd 7437 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑦𝑝𝑥) = (𝑦 + 𝑥))
2019eleq1d 2814 . . . . . . . . . . 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 3339 . . . . . . . . 9 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
2315, 22raleqbidv 3339 . . . . . . . 8 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
2410, 14, 23sbcied2 3824 . . . . . . 7 ((𝑔 = 𝐺𝑏 = 𝑋) → ([(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
256, 9, 24sbcied2 3824 . . . . . 6 (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
265, 25rabeqbidv 3446 . . . . 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 6910 . . . . . 6 (SubGrp‘𝐺) ∈ V
2827rabex 5334 . . . . 5 {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ∈ V
2926, 1, 28fvmpt 7005 . . . 4 (𝐺 ∈ Grp → (NrmSGrp‘𝐺) = {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)})
3029eleq2d 2815 . . 3 (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}))
31 eleq2 2818 . . . . . 6 (𝑠 = 𝑆 → ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑆))
32 eleq2 2818 . . . . . 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 3215 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
3534elrab 3682 . . 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 205  wa 395   = wceq 1534  wcel 2099  wral 3058  {crab 3429  Vcvv 3471  [wsbc 3776  cfv 6548  (class class class)co 7420  Basecbs 17180  +gcplusg 17233  Grpcgrp 18890  SubGrpcsubg 19075  NrmSGrpcnsg 19076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-subg 19078  df-nsg 19079
This theorem is referenced by:  isnsg2  19111  nsgbi  19112  nsgsubg  19113  isnsg4  19122  nmznsg  19123  ablnsg  19802  rzgrp  21555
  Copyright terms: Public domain W3C validator