Theorem psgnfval 19296

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 6847	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (SymGrp‘𝑑) = (SymGrp‘𝐷))
3		psgnfval.g	. . . . . . . . . 10 ⊢ 𝐺 = (SymGrp‘𝐷)
4	2, 3	eqtr4di 2789	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (SymGrp‘𝑑) = 𝐺)
5	4	fveq2d 6851	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = (Base‘𝐺))
6		psgnfval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
7	5, 6	eqtr4di 2789	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = 𝐵)
8		rabeq 3419	. . . . . . 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 2789	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} = 𝐹)
12		fveq2 6847	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (pmTrsp‘𝑑) = (pmTrsp‘𝐷))
13	12	rneqd 5898	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = ran (pmTrsp‘𝐷))
14		psgnfval.t	. . . . . . . . 9 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
15	13, 14	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = 𝑇)
16		wrdeq 14436	. . . . . . . 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 7377	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ((SymGrp‘𝑑) Σg 𝑤) = (𝐺 Σg 𝑤))
19	18	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ↔ 𝑥 = (𝐺 Σg 𝑤)))
20	19	anbi1d 630	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ (𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
21	17, 20	rexeqbidv 3318	. . . . . 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 6485	. . . . 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 5201	. . . 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 19287	. . . 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 6861	. . . . . 6 ⊢ 𝐵 ∈ V
26	10, 25	rabex2 5296	. . . . 5 ⊢ 𝐹 ∈ V
27	26	mptex 7178	. . . 4 ⊢ (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) ∈ V
28	23, 24, 27	fvmpt 6953	. . 3 ⊢ (𝐷 ∈ V → (pmSgn‘𝐷) = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
29		fvprc 6839	. . . 4 ⊢ (¬ 𝐷 ∈ V → (pmSgn‘𝐷) = ∅)
30		fvprc 6839	. . . . . . . . . . . . 13 ⊢ (¬ 𝐷 ∈ V → (SymGrp‘𝐷) = ∅)
31	3, 30	eqtrid 2783	. . . . . . . . . . . 12 ⊢ (¬ 𝐷 ∈ V → 𝐺 = ∅)
32	31	fveq2d 6851	. . . . . . . . . . 11 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐺) = (Base‘∅))
33		base0 17099	. . . . . . . . . . 11 ⊢ ∅ = (Base‘∅)
34	32, 33	eqtr4di 2789	. . . . . . . . . 10 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐺) = ∅)
35	6, 34	eqtrid 2783	. . . . . . . . 9 ⊢ (¬ 𝐷 ∈ V → 𝐵 = ∅)
36		rabeq 3419	. . . . . . . . 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 4347	. . . . . . . 8 ⊢ {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅
39	37, 38	eqtrdi 2787	. . . . . . 7 ⊢ (¬ 𝐷 ∈ V → {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅)
40	10, 39	eqtrid 2783	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → 𝐹 = ∅)
41	40	mpteq1d 5205	. . . . 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 6648	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = ∅
43	41, 42	eqtrdi 2787	. . . 4 ⊢ (¬ 𝐷 ∈ V → (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = ∅)
44	29, 43	eqtr4d 2774	. . 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 2759	1 ⊢ 𝑁 = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3069  {crab 3405  Vcvv 3446   ∖ cdif 3910  ∅c0 4287   ↦ cmpt 5193   I cid 5535  dom cdm 5638  ran crn 5639  ℩cio 6451  ‘cfv 6501  (class class class)co 7362  Fincfn 8890  1c1 11061  -cneg 11395  ↑cexp 13977  ♯chash 14240  Word cword 14414  Basecbs 17094   Σ_g cgsu 17336  SymGrpcsymg 19162  pmTrspcpmtr 19237  pmSgncpsgn 19285

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  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 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-n0 12423  df-z 12509  df-uz 12773  df-fz 13435  df-fzo 13578  df-hash 14241  df-word 14415  df-slot 17065  df-ndx 17077  df-base 17095  df-psgn 19287

This theorem is referenced by:  psgnfn  19297  psgnval  19303  psgnfvalfi  19309