Theorem psgnfval 19523

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 6863	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (SymGrp‘𝑑) = (SymGrp‘𝐷))
3		psgnfval.g	. . . . . . . . . 10 ⊢ 𝐺 = (SymGrp‘𝐷)
4	2, 3	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → (SymGrp‘𝑑) = 𝐺)
5	4	fveq2d 6867	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = (Base‘𝐺))
6		psgnfval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
7	5, 6	eqtr4di 2814	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (Base‘(SymGrp‘𝑑)) = 𝐵)
8		rabeq 3427	. . . . . . 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 2814	. . . . 5 ⊢ (𝑑 = 𝐷 → {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} = 𝐹)
12		fveq2 6863	. . . . . . . . . 10 ⊢ (𝑑 = 𝐷 → (pmTrsp‘𝑑) = (pmTrsp‘𝐷))
13	12	rneqd 5912	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = ran (pmTrsp‘𝐷))
14		psgnfval.t	. . . . . . . . 9 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
15	13, 14	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → ran (pmTrsp‘𝑑) = 𝑇)
16		wrdeq 14546	. . . . . . . 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 7407	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ((SymGrp‘𝑑) Σg 𝑤) = (𝐺 Σg 𝑤))
19	18	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ↔ 𝑥 = (𝐺 Σg 𝑤)))
20	19	anbi1d 640	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ (𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
21	17, 20	rexeqbidv 3336	. . . . . 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 6501	. . . . 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 5186	. . . 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 19514	. . . 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 6877	. . . . . 6 ⊢ 𝐵 ∈ V
26	10, 25	rabex2 5296	. . . . 5 ⊢ 𝐹 ∈ V
27	26	mptex 7203	. . . 4 ⊢ (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) ∈ V
28	23, 24, 27	fvmpt 6971	. . 3 ⊢ (𝐷 ∈ V → (pmSgn‘𝐷) = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
29		fvprc 6855	. . . 4 ⊢ (¬ 𝐷 ∈ V → (pmSgn‘𝐷) = ∅)
30		fvprc 6855	. . . . . . . . . . . . 13 ⊢ (¬ 𝐷 ∈ V → (SymGrp‘𝐷) = ∅)
31	3, 30	eqtrid 2808	. . . . . . . . . . . 12 ⊢ (¬ 𝐷 ∈ V → 𝐺 = ∅)
32	31	fveq2d 6867	. . . . . . . . . . 11 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐺) = (Base‘∅))
33		base0 17233	. . . . . . . . . . 11 ⊢ ∅ = (Base‘∅)
34	32, 33	eqtr4di 2814	. . . . . . . . . 10 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐺) = ∅)
35	6, 34	eqtrid 2808	. . . . . . . . 9 ⊢ (¬ 𝐷 ∈ V → 𝐵 = ∅)
36		rabeq 3427	. . . . . . . . 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 4338	. . . . . . . 8 ⊢ {𝑝 ∈ ∅ ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅
39	37, 38	eqtrdi 2812	. . . . . . 7 ⊢ (¬ 𝐷 ∈ V → {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = ∅)
40	10, 39	eqtrid 2808	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → 𝐹 = ∅)
41	40	mpteq1d 5189	. . . . 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 6659	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = ∅
43	41, 42	eqtrdi 2812	. . . 4 ⊢ (¬ 𝐷 ∈ V → (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = ∅)
44	29, 43	eqtr4d 2799	. . 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 2784	1 ⊢ 𝑁 = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  {crab 3413  Vcvv 3453   ∖ cdif 3901  ∅c0 4285   ↦ cmpt 5180   I cid 5539  dom cdm 5645  ran crn 5646  ℩cio 6471  ‘cfv 6517  (class class class)co 7392  Fincfn 8923  1c1 11071  -cneg 11412  ↑cexp 14071  ♯chash 14340  Word cword 14523  Basecbs 17228   Σ_g cgsu 17452  SymGrpcsymg 19392  pmTrspcpmtr 19464  pmSgncpsgn 19512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-n0 12479  df-z 12566  df-uz 12837  df-fz 13510  df-fzo 13657  df-hash 14341  df-word 14524  df-slot 17201  df-ndx 17213  df-base 17229  df-psgn 19514

This theorem is referenced by:  psgnfn  19524  psgnval  19530  psgnfvalfi  19536