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Theorem psgnfvalfi 19419

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 19406	. 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 19418	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐵) → dom (𝑝 ∖ I ) ∈ Fin)
8	7	ralrimiva 3125	. . . . 5 ⊢ (𝐷 ∈ Fin → ∀𝑝 ∈ 𝐵 dom (𝑝 ∖ I ) ∈ Fin)
9		rabid2 3436	. . . . 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 5192	. 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 3402 ∖ cdif 3908 ↦ cmpt 5183 I cid 5525 dom cdm 5631 ran crn 5632 ℩cio 6450 ‘cfv 6499 (class class class)co 7369 Fincfn 8895 1c1 11045 -cneg 11382 ↑cexp 14002 ♯chash 14271 Word cword 14454 Basecbs 17155 Σ_g cgsu 17379 SymGrpcsymg 19275 pmTrspcpmtr 19347 pmSgncpsgn 19395

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

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 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-fzo 13592 df-hash 14272 df-word 14455 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-tset 17215 df-efmnd 18772 df-symg 19276 df-psgn 19397

This theorem is referenced by: (None)

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