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Theorem isnsg 17544
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 17513 . . . 4 NrmSGrp = (𝑔 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑔) ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
21dmmptss 5590 . . 3 dom NrmSGrp ⊆ Grp
3 elfvdm 6177 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ dom NrmSGrp)
42, 3sseldi 3581 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp)
5 subgrcl 17520 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
65adantr 481 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)) → 𝐺 ∈ Grp)
7 fveq2 6148 . . . . . 6 (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
8 fvex 6158 . . . . . . . 8 (Base‘𝑔) ∈ V
98a1i 11 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) ∈ V)
10 fveq2 6148 . . . . . . . 8 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
11 isnsg.1 . . . . . . . 8 𝑋 = (Base‘𝐺)
1210, 11syl6eqr 2673 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
13 fvex 6158 . . . . . . . . 9 (+g𝑔) ∈ V
1413a1i 11 . . . . . . . 8 ((𝑔 = 𝐺𝑏 = 𝑋) → (+g𝑔) ∈ V)
15 simpl 473 . . . . . . . . . 10 ((𝑔 = 𝐺𝑏 = 𝑋) → 𝑔 = 𝐺)
1615fveq2d 6152 . . . . . . . . 9 ((𝑔 = 𝐺𝑏 = 𝑋) → (+g𝑔) = (+g𝐺))
17 isnsg.2 . . . . . . . . 9 + = (+g𝐺)
1816, 17syl6eqr 2673 . . . . . . . 8 ((𝑔 = 𝐺𝑏 = 𝑋) → (+g𝑔) = + )
19 simplr 791 . . . . . . . . 9 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑏 = 𝑋)
20 simpr 477 . . . . . . . . . . . . 13 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑝 = + )
2120oveqd 6621 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
2221eleq1d 2683 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑠))
2320oveqd 6621 . . . . . . . . . . . 12 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑦𝑝𝑥) = (𝑦 + 𝑥))
2423eleq1d 2683 . . . . . . . . . . 11 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑦𝑝𝑥) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠))
2522, 24bibi12d 335 . . . . . . . . . 10 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
2619, 25raleqbidv 3141 . . . . . . . . 9 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
2719, 26raleqbidv 3141 . . . . . . . 8 (((𝑔 = 𝐺𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
2814, 18, 27sbcied2 3455 . . . . . . 7 ((𝑔 = 𝐺𝑏 = 𝑋) → ([(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
299, 12, 28sbcied2 3455 . . . . . 6 (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑥𝑏𝑦𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
307, 29rabeqbidv 3181 . . . . 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)})
31 fvex 6158 . . . . . 6 (SubGrp‘𝐺) ∈ V
3231rabex 4773 . . . . 5 {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ∈ V
3330, 1, 32fvmpt 6239 . . . 4 (𝐺 ∈ Grp → (NrmSGrp‘𝐺) = {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)})
3433eleq2d 2684 . . 3 (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}))
35 eleq2 2687 . . . . . 6 (𝑠 = 𝑆 → ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑆))
36 eleq2 2687 . . . . . 6 (𝑠 = 𝑆 → ((𝑦 + 𝑥) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑆))
3735, 36bibi12d 335 . . . . 5 (𝑠 = 𝑆 → (((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠) ↔ ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
38372ralbidv 2983 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠) ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
3938elrab 3346 . . 3 (𝑆 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
4034, 39syl6bb 276 . 2 (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))))
414, 6, 40pm5.21nii 368 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 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  [wsbc 3417  dom cdm 5074  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Grpcgrp 17343  SubGrpcsubg 17509  NrmSGrpcnsg 17510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-subg 17512  df-nsg 17513
This theorem is referenced by:  isnsg2  17545  nsgbi  17546  nsgsubg  17547  isnsg4  17558  nmznsg  17559  ablnsg  18171  rzgrp  24204
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