Theorem psgnfval 18620

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 = {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}
psgnfval.t	⊢ 𝑇 = ran (pmTrsp‘𝐷)
psgnfval.n	⊢ 𝑁 = (pmSgn‘𝐷)

Assertion

Ref	Expression
psgnfval	⊢ 𝑁 = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ 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.

Step	Hyp	Ref	Expression
1		psgnfval.n	. 2 ⊢ 𝑁 = (pmSgn‘𝐷)
2		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (SymGrp‘𝑑) = (SymGrp‘𝐷))
3		psgnfval.g	. . . . . . . . . 10 ⊢ 𝐺 = (SymGrp‘𝐷)
4	2, 3	eqtr4di 2851	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (SymGrp‘𝑑) = 𝐺)
5	4	fveq2d 6649	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = (Base‘𝐺))
6		psgnfval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
7	5, 6	eqtr4di 2851	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = 𝐵)
8		rabeq 3431	. . . . . . 7 ⊢ ((Base‘(SymGrp‘𝑑)) = 𝐵 → {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin})
9	7, 8	syl 17	. . . . . 6 ⊢ (𝑑 = 𝐷 → {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin})
10		psgnfval.f	. . . . . 6 ⊢ 𝐹 = {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}
11	9, 10	eqtr4di 2851	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} = 𝐹)
12		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (pmTrsp‘𝑑) = (pmTrsp‘𝐷))
13	12	rneqd 5772	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = ran (pmTrsp‘𝐷))
14		psgnfval.t	. . . . . . . . 9 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
15	13, 14	eqtr4di 2851	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = 𝑇)
16		wrdeq 13879	. . . . . . . 8 ⊢ (ran (pmTrsp‘𝑑) = 𝑇 → Word ran (pmTrsp‘𝑑) = Word 𝑇)
17	15, 16	syl 17	. . . . . . 7 ⊢ (𝑑 = 𝐷 → Word ran (pmTrsp‘𝑑) = Word 𝑇)
18	4	oveq1d 7150	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ((SymGrp‘𝑑) Σg 𝑤) = (𝐺 Σg 𝑤))
19	18	eqeq2d 2809	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ↔ 𝑥 = (𝐺 Σg 𝑤)))
20	19	anbi1d 632	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ (𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
21	17, 20	rexeqbidv 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)))))
22	21	iotabidv 6308	. . . . 5 ⊢ (𝑑 = 𝐷 → (℩𝑠∃𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
23	11, 22	mpteq12dv 5115	. . . 4 ⊢ (𝑑 = 𝐷 → (𝑥 ∈ {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦ (℩𝑠∃𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
24		df-psgn 18611	. . . 4 ⊢ pmSgn = (𝑑 ∈ V ↦ (𝑥 ∈ {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦ (℩𝑠∃𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
25	6	fvexi 6659	. . . . . 6 ⊢ 𝐵 ∈ V
26	10, 25	rabex2 5201	. . . . 5 ⊢ 𝐹 ∈ V
27	26	mptex 6963	. . . 4 ⊢ (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) ∈ V
28	23, 24, 27	fvmpt 6745	. . 3 ⊢ (𝐷 ∈ V → (pmSgn‘𝐷) = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
29		fvprc 6638	. . . 4 ⊢ (¬ 𝐷 ∈ V → (pmSgn‘𝐷) = ∅)
30		fvprc 6638	. . . . . . . . . . . . 13 ⊢ (¬ 𝐷 ∈ V → (SymGrp‘𝐷) = ∅)
31	3, 30	syl5eq 2845	. . . . . . . . . . . 12 ⊢ (¬ 𝐷 ∈ V → 𝐺 = ∅)
32	31	fveq2d 6649	. . . . . . . . . . 11 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐺) = (Base‘∅))
33		base0 16528	. . . . . . . . . . 11 ⊢ ∅ = (Base‘∅)
34	32, 33	eqtr4di 2851	. . . . . . . . . 10 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐺) = ∅)
35	6, 34	syl5eq 2845	. . . . . . . . 9 ⊢ (¬ 𝐷 ∈ V → 𝐵 = ∅)
36		rabeq 3431	. . . . . . . . 9 ⊢ (𝐵 = ∅ → {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin})
37	35, 36	syl 17	. . . . . . . 8 ⊢ (¬ 𝐷 ∈ V → {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin})
38		rab0 4291	. . . . . . . 8 ⊢ {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅
39	37, 38	eqtrdi 2849	. . . . . . 7 ⊢ (¬ 𝐷 ∈ V → {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅)
40	10, 39	syl5eq 2845	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → 𝐹 = ∅)
41	40	mpteq1d 5119	. . . . 5 ⊢ (¬ 𝐷 ∈ V → (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = (𝑥 ∈ ∅ ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
42		mpt0 6462	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = ∅
43	41, 42	eqtrdi 2849	. . . 4 ⊢ (¬ 𝐷 ∈ V → (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = ∅)
44	29, 43	eqtr4d 2836	. . 3 ⊢ (¬ 𝐷 ∈ V → (pmSgn‘𝐷) = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
45	28, 44	pm2.61i 185	. 2 ⊢ (pmSgn‘𝐷) = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
46	1, 45	eqtri 2821	1 ⊢ 𝑁 = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  {crab 3110  Vcvv 3441   ∖ cdif 3878  ∅c0 4243   ↦ cmpt 5110   I cid 5424  dom cdm 5519  ran crn 5520  ℩cio 6281  ‘cfv 6324  (class class class)co 7135  Fincfn 8492  1c1 10527  -cneg 10860  ↑cexp 13425  ♯chash 13686  Word cword 13857  Basecbs 16475   Σ_g cgsu 16706  SymGrpcsymg 18487  pmTrspcpmtr 18561  pmSgncpsgn 18609

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 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

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 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-slot 16479  df-base 16481  df-psgn 18611

This theorem is referenced by:  psgnfn  18621  psgnval  18627  psgnfvalfi  18633