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Theorem psgnfvalfi 19392

Description: Function definition of the permutation sign function for permutations of finite sets. (Contributed by AV, 13-Jan-2019.)

**Hypotheses**
Ref	Expression
psgnfvalfi.g	⊢ 𝐺 = (SymGrp‘𝐷)
psgnfvalfi.b	⊢ 𝐵 = (Base‘𝐺)
psgnfvalfi.t	⊢ 𝑇 = ran (pmTrsp‘𝐷)
psgnfvalfi.n	⊢ 𝑁 = (pmSgn‘𝐷)

**Assertion**
Ref	Expression
psgnfvalfi	⊢ (𝐷 ∈ Fin → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σ_g 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))

Distinct variable groups: 𝑥,𝐷,𝑠,𝑤 𝑥,𝐵 𝐷,𝑠,𝑤 𝑤,𝑇 Allowed substitution hints: 𝐵(𝑤,𝑠) 𝑇(𝑥,𝑠) 𝐺(𝑥,𝑤,𝑠) 𝑁(𝑥,𝑤,𝑠)

Proof of Theorem psgnfvalfi Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psgnfvalfi.g	. . 3 ⊢ 𝐺 = (SymGrp‘𝐷)
2		psgnfvalfi.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2729	. . 3 ⊢ {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}
4		psgnfvalfi.t	. . 3 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
5		psgnfvalfi.n	. . 3 ⊢ 𝑁 = (pmSgn‘𝐷)
6	1, 2, 3, 4, 5	psgnfval 19379	. 2 ⊢ 𝑁 = (𝑥 ∈ {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σ_g 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
7	1, 2	sygbasnfpfi 19391	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐵) → dom (𝑝 ∖ I ) ∈ Fin)
8	7	ralrimiva 3121	. . . . 5 ⊢ (𝐷 ∈ Fin → ∀𝑝 ∈ 𝐵 dom (𝑝 ∖ I ) ∈ Fin)
9		rabid2 3428	. . . . 5 ⊢ (𝐵 = {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} ↔ ∀𝑝 ∈ 𝐵 dom (𝑝 ∖ I ) ∈ Fin)
10	8, 9	sylibr 234	. . . 4 ⊢ (𝐷 ∈ Fin → 𝐵 = {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin})
11	10	eqcomd 2735	. . 3 ⊢ (𝐷 ∈ Fin → {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = 𝐵)
12	11	mpteq1d 5182	. 2 ⊢ (𝐷 ∈ Fin → (𝑥 ∈ {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σ_g 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) = (𝑥 ∈ 𝐵 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σ_g 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
13	6, 12	eqtrid 2776	1 ⊢ (𝐷 ∈ Fin → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σ_g 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 {crab 3394 ∖ cdif 3900 ↦ cmpt 5173 I cid 5513 dom cdm 5619 ran crn 5620 ℩cio 6436 ‘cfv 6482 (class class class)co 7349 Fincfn 8872 1c1 11010 -cneg 11348 ↑cexp 13968 ♯chash 14237 Word cword 14420 Basecbs 17120 Σ_g cgsu 17344 SymGrpcsymg 19248 pmTrspcpmtr 19320 pmSgncpsgn 19368

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

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 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-uz 12736 df-fz 13411 df-fzo 13558 df-hash 14238 df-word 14421 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-tset 17180 df-efmnd 18743 df-symg 19249 df-psgn 19370

This theorem is referenced by: (None)

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