Theorem psgnfval 19430

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 6858	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (SymGrp‘𝑑) = (SymGrp‘𝐷))
3		psgnfval.g	. . . . . . . . . 10 ⊢ 𝐺 = (SymGrp‘𝐷)
4	2, 3	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (SymGrp‘𝑑) = 𝐺)
5	4	fveq2d 6862	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = (Base‘𝐺))
6		psgnfval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
7	5, 6	eqtr4di 2782	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = 𝐵)
8		rabeq 3420	. . . . . . 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 2782	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} = 𝐹)
12		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (pmTrsp‘𝑑) = (pmTrsp‘𝐷))
13	12	rneqd 5902	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = ran (pmTrsp‘𝐷))
14		psgnfval.t	. . . . . . . . 9 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
15	13, 14	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = 𝑇)
16		wrdeq 14501	. . . . . . . 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 7402	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ((SymGrp‘𝑑) Σg 𝑤) = (𝐺 Σg 𝑤))
19	18	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ↔ 𝑥 = (𝐺 Σg 𝑤)))
20	19	anbi1d 631	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ (𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
21	17, 20	rexeqbidv 3320	. . . . . 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 6495	. . . . 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 5194	. . . 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 19421	. . . 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 6872	. . . . . 6 ⊢ 𝐵 ∈ V
26	10, 25	rabex2 5296	. . . . 5 ⊢ 𝐹 ∈ V
27	26	mptex 7197	. . . 4 ⊢ (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) ∈ V
28	23, 24, 27	fvmpt 6968	. . 3 ⊢ (𝐷 ∈ V → (pmSgn‘𝐷) = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
29		fvprc 6850	. . . 4 ⊢ (¬ 𝐷 ∈ V → (pmSgn‘𝐷) = ∅)
30		fvprc 6850	. . . . . . . . . . . . 13 ⊢ (¬ 𝐷 ∈ V → (SymGrp‘𝐷) = ∅)
31	3, 30	eqtrid 2776	. . . . . . . . . . . 12 ⊢ (¬ 𝐷 ∈ V → 𝐺 = ∅)
32	31	fveq2d 6862	. . . . . . . . . . 11 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐺) = (Base‘∅))
33		base0 17184	. . . . . . . . . . 11 ⊢ ∅ = (Base‘∅)
34	32, 33	eqtr4di 2782	. . . . . . . . . 10 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐺) = ∅)
35	6, 34	eqtrid 2776	. . . . . . . . 9 ⊢ (¬ 𝐷 ∈ V → 𝐵 = ∅)
36		rabeq 3420	. . . . . . . . 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 4349	. . . . . . . 8 ⊢ {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅
39	37, 38	eqtrdi 2780	. . . . . . 7 ⊢ (¬ 𝐷 ∈ V → {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅)
40	10, 39	eqtrid 2776	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → 𝐹 = ∅)
41	40	mpteq1d 5197	. . . . 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 6660	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = ∅
43	41, 42	eqtrdi 2780	. . . 4 ⊢ (¬ 𝐷 ∈ V → (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = ∅)
44	29, 43	eqtr4d 2767	. . 3 ⊢ (¬ 𝐷 ∈ V → (pmSgn‘𝐷) = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
45	28, 44	pm2.61i 182	. 2 ⊢ (pmSgn‘𝐷) = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
46	1, 45	eqtri 2752	1 ⊢ 𝑁 = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3405  Vcvv 3447   ∖ cdif 3911  ∅c0 4296   ↦ cmpt 5188   I cid 5532  dom cdm 5638  ran crn 5639  ℩cio 6462  ‘cfv 6511  (class class class)co 7387  Fincfn 8918  1c1 11069  -cneg 11406  ↑cexp 14026  ♯chash 14295  Word cword 14478  Basecbs 17179   Σ_g cgsu 17403  SymGrpcsymg 19299  pmTrspcpmtr 19371  pmSgncpsgn 19419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-hash 14296  df-word 14479  df-slot 17152  df-ndx 17164  df-base 17180  df-psgn 19421

This theorem is referenced by:  psgnfn  19431  psgnval  19437  psgnfvalfi  19443