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Theorem isnsg 12993
Description: Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
isnsg.1 𝑋 = (Baseβ€˜πΊ)
isnsg.2 + = (+gβ€˜πΊ)
Assertion
Ref Expression
isnsg (𝑆 ∈ (NrmSGrpβ€˜πΊ) ↔ (𝑆 ∈ (SubGrpβ€˜πΊ) ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑆 ↔ (𝑦 + π‘₯) ∈ 𝑆)))
Distinct variable groups:   π‘₯,𝑦,𝐺   π‘₯, + ,𝑦   π‘₯,𝑆,𝑦   π‘₯,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 12962 . . 3 NrmSGrp = (𝑔 ∈ Grp ↦ {𝑠 ∈ (SubGrpβ€˜π‘”) ∣ [(Baseβ€˜π‘”) / 𝑏][(+gβ€˜π‘”) / 𝑝]βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 ((π‘₯𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝π‘₯) ∈ 𝑠)})
21mptrcl 5597 . 2 (𝑆 ∈ (NrmSGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
3 subgrcl 12970 . . 3 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
43adantr 276 . 2 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑆 ↔ (𝑦 + π‘₯) ∈ 𝑆)) β†’ 𝐺 ∈ Grp)
5 fveq2 5514 . . . . . 6 (𝑔 = 𝐺 β†’ (SubGrpβ€˜π‘”) = (SubGrpβ€˜πΊ))
6 basfn 12512 . . . . . . . . . 10 Base Fn V
7 funfvex 5531 . . . . . . . . . . 11 ((Fun Base ∧ 𝑔 ∈ dom Base) β†’ (Baseβ€˜π‘”) ∈ V)
87funfni 5315 . . . . . . . . . 10 ((Base Fn V ∧ 𝑔 ∈ V) β†’ (Baseβ€˜π‘”) ∈ V)
96, 8mpan 424 . . . . . . . . 9 (𝑔 ∈ V β†’ (Baseβ€˜π‘”) ∈ V)
109elv 2741 . . . . . . . 8 (Baseβ€˜π‘”) ∈ V
1110a1i 9 . . . . . . 7 (𝑔 = 𝐺 β†’ (Baseβ€˜π‘”) ∈ V)
12 fveq2 5514 . . . . . . . 8 (𝑔 = 𝐺 β†’ (Baseβ€˜π‘”) = (Baseβ€˜πΊ))
13 isnsg.1 . . . . . . . 8 𝑋 = (Baseβ€˜πΊ)
1412, 13eqtr4di 2228 . . . . . . 7 (𝑔 = 𝐺 β†’ (Baseβ€˜π‘”) = 𝑋)
15 plusgslid 12563 . . . . . . . . . . 11 (+g = Slot (+gβ€˜ndx) ∧ (+gβ€˜ndx) ∈ β„•)
1615slotex 12481 . . . . . . . . . 10 (𝑔 ∈ V β†’ (+gβ€˜π‘”) ∈ V)
1716elv 2741 . . . . . . . . 9 (+gβ€˜π‘”) ∈ V
1817a1i 9 . . . . . . . 8 ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) β†’ (+gβ€˜π‘”) ∈ V)
19 simpl 109 . . . . . . . . . 10 ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) β†’ 𝑔 = 𝐺)
2019fveq2d 5518 . . . . . . . . 9 ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) β†’ (+gβ€˜π‘”) = (+gβ€˜πΊ))
21 isnsg.2 . . . . . . . . 9 + = (+gβ€˜πΊ)
2220, 21eqtr4di 2228 . . . . . . . 8 ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) β†’ (+gβ€˜π‘”) = + )
23 simplr 528 . . . . . . . . 9 (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) β†’ 𝑏 = 𝑋)
24 simpr 110 . . . . . . . . . . . . 13 (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) β†’ 𝑝 = + )
2524oveqd 5889 . . . . . . . . . . . 12 (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) β†’ (π‘₯𝑝𝑦) = (π‘₯ + 𝑦))
2625eleq1d 2246 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) β†’ ((π‘₯𝑝𝑦) ∈ 𝑠 ↔ (π‘₯ + 𝑦) ∈ 𝑠))
2724oveqd 5889 . . . . . . . . . . . 12 (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) β†’ (𝑦𝑝π‘₯) = (𝑦 + π‘₯))
2827eleq1d 2246 . . . . . . . . . . 11 (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) β†’ ((𝑦𝑝π‘₯) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠))
2926, 28bibi12d 235 . . . . . . . . . 10 (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) β†’ (((π‘₯𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝π‘₯) ∈ 𝑠) ↔ ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠)))
3023, 29raleqbidv 2684 . . . . . . . . 9 (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) β†’ (βˆ€π‘¦ ∈ 𝑏 ((π‘₯𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝π‘₯) ∈ 𝑠) ↔ βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠)))
3123, 30raleqbidv 2684 . . . . . . . 8 (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) β†’ (βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 ((π‘₯𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝π‘₯) ∈ 𝑠) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠)))
3218, 22, 31sbcied2 3000 . . . . . . 7 ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) β†’ ([(+gβ€˜π‘”) / 𝑝]βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 ((π‘₯𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝π‘₯) ∈ 𝑠) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠)))
3311, 14, 32sbcied2 3000 . . . . . 6 (𝑔 = 𝐺 β†’ ([(Baseβ€˜π‘”) / 𝑏][(+gβ€˜π‘”) / 𝑝]βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 ((π‘₯𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝π‘₯) ∈ 𝑠) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠)))
345, 33rabeqbidv 2732 . . . . 5 (𝑔 = 𝐺 β†’ {𝑠 ∈ (SubGrpβ€˜π‘”) ∣ [(Baseβ€˜π‘”) / 𝑏][(+gβ€˜π‘”) / 𝑝]βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 ((π‘₯𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝π‘₯) ∈ 𝑠)} = {𝑠 ∈ (SubGrpβ€˜πΊ) ∣ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠)})
35 id 19 . . . . 5 (𝐺 ∈ Grp β†’ 𝐺 ∈ Grp)
36 subgex 12967 . . . . . 6 (𝐺 ∈ Grp β†’ (SubGrpβ€˜πΊ) ∈ V)
37 rabexg 4145 . . . . . 6 ((SubGrpβ€˜πΊ) ∈ V β†’ {𝑠 ∈ (SubGrpβ€˜πΊ) ∣ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠)} ∈ V)
3836, 37syl 14 . . . . 5 (𝐺 ∈ Grp β†’ {𝑠 ∈ (SubGrpβ€˜πΊ) ∣ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠)} ∈ V)
391, 34, 35, 38fvmptd3 5608 . . . 4 (𝐺 ∈ Grp β†’ (NrmSGrpβ€˜πΊ) = {𝑠 ∈ (SubGrpβ€˜πΊ) ∣ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠)})
4039eleq2d 2247 . . 3 (𝐺 ∈ Grp β†’ (𝑆 ∈ (NrmSGrpβ€˜πΊ) ↔ 𝑆 ∈ {𝑠 ∈ (SubGrpβ€˜πΊ) ∣ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠)}))
41 eleq2 2241 . . . . . 6 (𝑠 = 𝑆 β†’ ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (π‘₯ + 𝑦) ∈ 𝑆))
42 eleq2 2241 . . . . . 6 (𝑠 = 𝑆 β†’ ((𝑦 + π‘₯) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑆))
4341, 42bibi12d 235 . . . . 5 (𝑠 = 𝑆 β†’ (((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠) ↔ ((π‘₯ + 𝑦) ∈ 𝑆 ↔ (𝑦 + π‘₯) ∈ 𝑆)))
44432ralbidv 2501 . . . 4 (𝑠 = 𝑆 β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑆 ↔ (𝑦 + π‘₯) ∈ 𝑆)))
4544elrab 2893 . . 3 (𝑆 ∈ {𝑠 ∈ (SubGrpβ€˜πΊ) ∣ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑠 ↔ (𝑦 + π‘₯) ∈ 𝑠)} ↔ (𝑆 ∈ (SubGrpβ€˜πΊ) ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑆 ↔ (𝑦 + π‘₯) ∈ 𝑆)))
4640, 45bitrdi 196 . 2 (𝐺 ∈ Grp β†’ (𝑆 ∈ (NrmSGrpβ€˜πΊ) ↔ (𝑆 ∈ (SubGrpβ€˜πΊ) ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑆 ↔ (𝑦 + π‘₯) ∈ 𝑆))))
472, 4, 46pm5.21nii 704 1 (𝑆 ∈ (NrmSGrpβ€˜πΊ) ↔ (𝑆 ∈ (SubGrpβ€˜πΊ) ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯ + 𝑦) ∈ 𝑆 ↔ (𝑦 + π‘₯) ∈ 𝑆)))
Colors of variables: wff set class
Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  {crab 2459  Vcvv 2737  [wsbc 2962   Fn wfn 5210  β€˜cfv 5215  (class class class)co 5872  Basecbs 12454  +gcplusg 12528  Grpcgrp 12809  SubGrpcsubg 12958  NrmSGrpcnsg 12959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-ov 5875  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-plusg 12541  df-subg 12961  df-nsg 12962
This theorem is referenced by:  isnsg2  12994  nsgbi  12995  nsgsubg  12996  isnsg4  13003  nmznsg  13004
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