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Theorem psgnfval 18624
 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 6646 . . . . . . . . . 10 (𝑑 = 𝐷 → (SymGrp‘𝑑) = (SymGrp‘𝐷))
3 psgnfval.g . . . . . . . . . 10 𝐺 = (SymGrp‘𝐷)
42, 3eqtr4di 2851 . . . . . . . . 9 (𝑑 = 𝐷 → (SymGrp‘𝑑) = 𝐺)
54fveq2d 6650 . . . . . . . 8 (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = (Base‘𝐺))
6 psgnfval.b . . . . . . . 8 𝐵 = (Base‘𝐺)
75, 6eqtr4di 2851 . . . . . . 7 (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = 𝐵)
8 rabeq 3431 . . . . . . 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, 10eqtr4di 2851 . . . . 5 (𝑑 = 𝐷 → {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} = 𝐹)
12 fveq2 6646 . . . . . . . . . 10 (𝑑 = 𝐷 → (pmTrsp‘𝑑) = (pmTrsp‘𝐷))
1312rneqd 5773 . . . . . . . . 9 (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = ran (pmTrsp‘𝐷))
14 psgnfval.t . . . . . . . . 9 𝑇 = ran (pmTrsp‘𝐷)
1513, 14eqtr4di 2851 . . . . . . . 8 (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = 𝑇)
16 wrdeq 13882 . . . . . . . 8 (ran (pmTrsp‘𝑑) = 𝑇 → Word ran (pmTrsp‘𝑑) = Word 𝑇)
1715, 16syl 17 . . . . . . 7 (𝑑 = 𝐷 → Word ran (pmTrsp‘𝑑) = Word 𝑇)
184oveq1d 7151 . . . . . . . . 9 (𝑑 = 𝐷 → ((SymGrp‘𝑑) Σg 𝑤) = (𝐺 Σg 𝑤))
1918eqeq2d 2809 . . . . . . . 8 (𝑑 = 𝐷 → (𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ↔ 𝑥 = (𝐺 Σg 𝑤)))
2019anbi1d 632 . . . . . . 7 (𝑑 = 𝐷 → ((𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ (𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
2117, 20rexeqbidv 3355 . . . . . 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 6309 . . . . 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 5116 . . . 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 18615 . . . 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))))))
256fvexi 6660 . . . . . 6 𝐵 ∈ V
2610, 25rabex2 5202 . . . . 5 𝐹 ∈ V
2726mptex 6964 . . . 4 (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) ∈ V
2823, 24, 27fvmpt 6746 . . 3 (𝐷 ∈ V → (pmSgn‘𝐷) = (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
29 fvprc 6639 . . . 4 𝐷 ∈ V → (pmSgn‘𝐷) = ∅)
30 fvprc 6639 . . . . . . . . . . . . 13 𝐷 ∈ V → (SymGrp‘𝐷) = ∅)
313, 30syl5eq 2845 . . . . . . . . . . . 12 𝐷 ∈ V → 𝐺 = ∅)
3231fveq2d 6650 . . . . . . . . . . 11 𝐷 ∈ V → (Base‘𝐺) = (Base‘∅))
33 base0 16531 . . . . . . . . . . 11 ∅ = (Base‘∅)
3432, 33eqtr4di 2851 . . . . . . . . . 10 𝐷 ∈ V → (Base‘𝐺) = ∅)
356, 34syl5eq 2845 . . . . . . . . 9 𝐷 ∈ V → 𝐵 = ∅)
36 rabeq 3431 . . . . . . . . 9 (𝐵 = ∅ → {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin})
3735, 36syl 17 . . . . . . . 8 𝐷 ∈ V → {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin})
38 rab0 4291 . . . . . . . 8 {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅
3937, 38eqtrdi 2849 . . . . . . 7 𝐷 ∈ V → {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅)
4010, 39syl5eq 2845 . . . . . 6 𝐷 ∈ V → 𝐹 = ∅)
4140mpteq1d 5120 . . . . 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))))))
42 mpt0 6463 . . . . 5 (𝑥 ∈ ∅ ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = ∅
4341, 42eqtrdi 2849 . . . 4 𝐷 ∈ V → (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = ∅)
4429, 43eqtr4d 2836 . . 3 𝐷 ∈ V → (pmSgn‘𝐷) = (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
4528, 44pm2.61i 185 . 2 (pmSgn‘𝐷) = (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
461, 45eqtri 2821 1 𝑁 = (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  {crab 3110  Vcvv 3441   ∖ cdif 3878  ∅c0 4243   ↦ cmpt 5111   I cid 5425  dom cdm 5520  ran crn 5521  ℩cio 6282  ‘cfv 6325  (class class class)co 7136  Fincfn 8495  1c1 10530  -cneg 10863  ↑cexp 13428  ♯chash 13689  Word cword 13860  Basecbs 16478   Σg cgsu 16709  SymGrpcsymg 18491  pmTrspcpmtr 18565  pmSgncpsgn 18613 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-n0 11889  df-z 11973  df-uz 12235  df-fz 12889  df-fzo 13032  df-hash 13690  df-word 13861  df-slot 16482  df-base 16484  df-psgn 18615 This theorem is referenced by:  psgnfn  18625  psgnval  18631  psgnfvalfi  18637
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