Theorem psgnfval 19473

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 6834	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (SymGrp‘𝑑) = (SymGrp‘𝐷))
3		psgnfval.g	. . . . . . . . . 10 ⊢ 𝐺 = (SymGrp‘𝐷)
4	2, 3	eqtr4di 2793	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (SymGrp‘𝑑) = 𝐺)
5	4	fveq2d 6838	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = (Base‘𝐺))
6		psgnfval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
7	5, 6	eqtr4di 2793	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = 𝐵)
8		rabeq 3406	. . . . . . 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 2793	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} = 𝐹)
12		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (pmTrsp‘𝑑) = (pmTrsp‘𝐷))
13	12	rneqd 5887	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = ran (pmTrsp‘𝐷))
14		psgnfval.t	. . . . . . . . 9 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
15	13, 14	eqtr4di 2793	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = 𝑇)
16		wrdeq 14496	. . . . . . . 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 7378	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ((SymGrp‘𝑑) Σg 𝑤) = (𝐺 Σg 𝑤))
19	18	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ↔ 𝑥 = (𝐺 Σg 𝑤)))
20	19	anbi1d 637	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ (𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
21	17, 20	rexeqbidv 3315	. . . . . 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 6476	. . . . 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 5166	. . . 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 19464	. . . 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 6848	. . . . . 6 ⊢ 𝐵 ∈ V
26	10, 25	rabex2 5276	. . . . 5 ⊢ 𝐹 ∈ V
27	26	mptex 7174	. . . 4 ⊢ (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) ∈ V
28	23, 24, 27	fvmpt 6942	. . 3 ⊢ (𝐷 ∈ V → (pmSgn‘𝐷) = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
29		fvprc 6826	. . . 4 ⊢ (¬ 𝐷 ∈ V → (pmSgn‘𝐷) = ∅)
30		fvprc 6826	. . . . . . . . . . . . 13 ⊢ (¬ 𝐷 ∈ V → (SymGrp‘𝐷) = ∅)
31	3, 30	eqtrid 2787	. . . . . . . . . . . 12 ⊢ (¬ 𝐷 ∈ V → 𝐺 = ∅)
32	31	fveq2d 6838	. . . . . . . . . . 11 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐺) = (Base‘∅))
33		base0 17182	. . . . . . . . . . 11 ⊢ ∅ = (Base‘∅)
34	32, 33	eqtr4di 2793	. . . . . . . . . 10 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐺) = ∅)
35	6, 34	eqtrid 2787	. . . . . . . . 9 ⊢ (¬ 𝐷 ∈ V → 𝐵 = ∅)
36		rabeq 3406	. . . . . . . . 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 4321	. . . . . . . 8 ⊢ {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅
39	37, 38	eqtrdi 2791	. . . . . . 7 ⊢ (¬ 𝐷 ∈ V → {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅)
40	10, 39	eqtrid 2787	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → 𝐹 = ∅)
41	40	mpteq1d 5169	. . . . 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 6634	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = ∅
43	41, 42	eqtrdi 2791	. . . 4 ⊢ (¬ 𝐷 ∈ V → (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = ∅)
44	29, 43	eqtr4d 2778	. . 3 ⊢ (¬ 𝐷 ∈ V → (pmSgn‘𝐷) = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
45	28, 44	pm2.61i 183	. 2 ⊢ (pmSgn‘𝐷) = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
46	1, 45	eqtri 2763	1 ⊢ 𝑁 = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3064  {crab 3392  Vcvv 3432   ∖ cdif 3887  ∅c0 4268   ↦ cmpt 5160   I cid 5519  dom cdm 5625  ran crn 5626  ℩cio 6446  ‘cfv 6492  (class class class)co 7363  Fincfn 8890  1c1 11037  -cneg 11376  ↑cexp 14021  ♯chash 14290  Word cword 14473  Basecbs 17177   Σ_g cgsu 17401  SymGrpcsymg 19342  pmTrspcpmtr 19414  pmSgncpsgn 19462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-fzo 13607  df-hash 14291  df-word 14474  df-slot 17150  df-ndx 17162  df-base 17178  df-psgn 19464

This theorem is referenced by:  psgnfn  19474  psgnval  19480  psgnfvalfi  19486