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Theorem psgnfval 17914
Description: Function definition of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
Hypotheses
Ref Expression
psgnfval.g 𝐺 = (SymGrp‘𝐷)
psgnfval.b 𝐵 = (Base‘𝐺)
psgnfval.f 𝐹 = {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}
psgnfval.t 𝑇 = ran (pmTrsp‘𝐷)
psgnfval.n 𝑁 = (pmSgn‘𝐷)
Assertion
Ref Expression
psgnfval 𝑁 = (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
Distinct variable groups:   𝑠,𝑝,𝑤,𝑥   𝐷,𝑠,𝑤,𝑥   𝑥,𝐹   𝑤,𝑇   𝐵,𝑝
Allowed substitution hints:   𝐵(𝑥,𝑤,𝑠)   𝐷(𝑝)   𝑇(𝑥,𝑠,𝑝)   𝐹(𝑤,𝑠,𝑝)   𝐺(𝑥,𝑤,𝑠,𝑝)   𝑁(𝑥,𝑤,𝑠,𝑝)

Proof of Theorem psgnfval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 psgnfval.n . 2 𝑁 = (pmSgn‘𝐷)
2 fveq2 6189 . . . . . . . . . 10 (𝑑 = 𝐷 → (SymGrp‘𝑑) = (SymGrp‘𝐷))
3 psgnfval.g . . . . . . . . . 10 𝐺 = (SymGrp‘𝐷)
42, 3syl6eqr 2673 . . . . . . . . 9 (𝑑 = 𝐷 → (SymGrp‘𝑑) = 𝐺)
54fveq2d 6193 . . . . . . . 8 (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = (Base‘𝐺))
6 psgnfval.b . . . . . . . 8 𝐵 = (Base‘𝐺)
75, 6syl6eqr 2673 . . . . . . 7 (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = 𝐵)
8 rabeq 3190 . . . . . . 7 ((Base‘(SymGrp‘𝑑)) = 𝐵 → {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin})
97, 8syl 17 . . . . . 6 (𝑑 = 𝐷 → {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin})
10 psgnfval.f . . . . . 6 𝐹 = {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}
119, 10syl6eqr 2673 . . . . 5 (𝑑 = 𝐷 → {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} = 𝐹)
12 fveq2 6189 . . . . . . . . . 10 (𝑑 = 𝐷 → (pmTrsp‘𝑑) = (pmTrsp‘𝐷))
1312rneqd 5351 . . . . . . . . 9 (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = ran (pmTrsp‘𝐷))
14 psgnfval.t . . . . . . . . 9 𝑇 = ran (pmTrsp‘𝐷)
1513, 14syl6eqr 2673 . . . . . . . 8 (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = 𝑇)
16 wrdeq 13322 . . . . . . . 8 (ran (pmTrsp‘𝑑) = 𝑇 → Word ran (pmTrsp‘𝑑) = Word 𝑇)
1715, 16syl 17 . . . . . . 7 (𝑑 = 𝐷 → Word ran (pmTrsp‘𝑑) = Word 𝑇)
184oveq1d 6662 . . . . . . . . 9 (𝑑 = 𝐷 → ((SymGrp‘𝑑) Σg 𝑤) = (𝐺 Σg 𝑤))
1918eqeq2d 2631 . . . . . . . 8 (𝑑 = 𝐷 → (𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ↔ 𝑥 = (𝐺 Σg 𝑤)))
2019anbi1d 741 . . . . . . 7 (𝑑 = 𝐷 → ((𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))) ↔ (𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
2117, 20rexeqbidv 3151 . . . . . 6 (𝑑 = 𝐷 → (∃𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))) ↔ ∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
2221iotabidv 5870 . . . . 5 (𝑑 = 𝐷 → (℩𝑠𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))) = (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
2311, 22mpteq12dv 4731 . . . 4 (𝑑 = 𝐷 → (𝑥 ∈ {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦ (℩𝑠𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))) = (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))))
24 df-psgn 17905 . . . 4 pmSgn = (𝑑 ∈ V ↦ (𝑥 ∈ {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦ (℩𝑠𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))))
25 fvex 6199 . . . . . . 7 (Base‘𝐺) ∈ V
266, 25eqeltri 2696 . . . . . 6 𝐵 ∈ V
2710, 26rabex2 4813 . . . . 5 𝐹 ∈ V
2827mptex 6483 . . . 4 (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))) ∈ V
2923, 24, 28fvmpt 6280 . . 3 (𝐷 ∈ V → (pmSgn‘𝐷) = (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))))
30 fvprc 6183 . . . 4 𝐷 ∈ V → (pmSgn‘𝐷) = ∅)
31 fvprc 6183 . . . . . . . . . . . . 13 𝐷 ∈ V → (SymGrp‘𝐷) = ∅)
323, 31syl5eq 2667 . . . . . . . . . . . 12 𝐷 ∈ V → 𝐺 = ∅)
3332fveq2d 6193 . . . . . . . . . . 11 𝐷 ∈ V → (Base‘𝐺) = (Base‘∅))
34 base0 15906 . . . . . . . . . . 11 ∅ = (Base‘∅)
3533, 34syl6eqr 2673 . . . . . . . . . 10 𝐷 ∈ V → (Base‘𝐺) = ∅)
366, 35syl5eq 2667 . . . . . . . . 9 𝐷 ∈ V → 𝐵 = ∅)
37 rabeq 3190 . . . . . . . . 9 (𝐵 = ∅ → {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin})
3836, 37syl 17 . . . . . . . 8 𝐷 ∈ V → {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin})
39 rab0 3953 . . . . . . . 8 {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅
4038, 39syl6eq 2671 . . . . . . 7 𝐷 ∈ V → {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅)
4110, 40syl5eq 2667 . . . . . 6 𝐷 ∈ V → 𝐹 = ∅)
4241mpteq1d 4736 . . . . 5 𝐷 ∈ V → (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))) = (𝑥 ∈ ∅ ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))))
43 mpt0 6019 . . . . 5 (𝑥 ∈ ∅ ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))) = ∅
4442, 43syl6eq 2671 . . . 4 𝐷 ∈ V → (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))) = ∅)
4530, 44eqtr4d 2658 . . 3 𝐷 ∈ V → (pmSgn‘𝐷) = (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))))
4629, 45pm2.61i 176 . 2 (pmSgn‘𝐷) = (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
471, 46eqtri 2643 1 𝑁 = (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1482  wcel 1989  wrex 2912  {crab 2915  Vcvv 3198  cdif 3569  c0 3913  cmpt 4727   I cid 5021  dom cdm 5112  ran crn 5113  cio 5847  cfv 5886  (class class class)co 6647  Fincfn 7952  1c1 9934  -cneg 10264  cexp 12855  #chash 13112  Word cword 13286  Basecbs 15851   Σg cgsu 16095  SymGrpcsymg 17791  pmTrspcpmtr 17855  pmSgncpsgn 17903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-card 8762  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-n0 11290  df-z 11375  df-uz 11685  df-fz 12324  df-fzo 12462  df-hash 13113  df-word 13294  df-slot 15855  df-base 15857  df-psgn 17905
This theorem is referenced by:  psgnfn  17915  psgnval  17921  psgnfvalfi  17927
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