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Theorem psgnfvalfi 19483

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 2737	. . 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 19470	. 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 19482	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐵) → dom (𝑝 ∖ I ) ∈ Fin)
8	7	ralrimiva 3130	. . . . 5 ⊢ (𝐷 ∈ Fin → ∀𝑝 ∈ 𝐵 dom (𝑝 ∖ I ) ∈ Fin)
9		rabid2 3423	. . . . 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 2743	. . 3 ⊢ (𝐷 ∈ Fin → {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin} = 𝐵)
12	11	mpteq1d 5176	. 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 2784	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 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 {crab 3390 ∖ cdif 3887 ↦ cmpt 5167 I cid 5520 dom cdm 5626 ran crn 5627 ℩cio 6448 ‘cfv 6494 (class class class)co 7362 Fincfn 8888 1c1 11034 -cneg 11373 ↑cexp 14018 ♯chash 14287 Word cword 14470 Basecbs 17174 Σ_g cgsu 17398 SymGrpcsymg 19339 pmTrspcpmtr 19411 pmSgncpsgn 19459

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-er 8638 df-map 8770 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-uz 12784 df-fz 13457 df-fzo 13604 df-hash 14288 df-word 14471 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-tset 17234 df-efmnd 18832 df-symg 19340 df-psgn 19461

This theorem is referenced by: (None)

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