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Theorem psgnsn 19507

Description: The permutation sign function for a singleton. (Contributed by AV, 6-Aug-2019.)

**Hypotheses**
Ref	Expression
psgnsn.0	⊢ 𝐷 = {𝐴}
psgnsn.g	⊢ 𝐺 = (SymGrp‘𝐷)
psgnsn.b	⊢ 𝐵 = (Base‘𝐺)
psgnsn.n	⊢ 𝑁 = (pmSgn‘𝐷)

**Assertion**
Ref	Expression
psgnsn	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1)

**Proof of Theorem psgnsn**
Step	Hyp	Ref	Expression
1		eqid 2734	. . . . 5 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
2	1	gsum0 18667	. . . 4 ⊢ (𝐺 Σ_g ∅) = (0_g‘𝐺)
3		psgnsn.g	. . . . . . . 8 ⊢ 𝐺 = (SymGrp‘𝐷)
4		psgnsn.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
5		psgnsn.0	. . . . . . . 8 ⊢ 𝐷 = {𝐴}
6	3, 4, 5	symg1bas 19377	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐵 = {{⟨𝐴, 𝐴⟩}})
7	6	eleq2d 2819	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {{⟨𝐴, 𝐴⟩}}))
8	7	biimpa 476	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ {{⟨𝐴, 𝐴⟩}})
9		elsni 4623	. . . . . 6 ⊢ (𝑋 ∈ {{⟨𝐴, 𝐴⟩}} → 𝑋 = {⟨𝐴, 𝐴⟩})
10	5	reseq2i 5974	. . . . . . . . . 10 ⊢ ( I ↾ 𝐷) = ( I ↾ {𝐴})
11		snex 5416	. . . . . . . . . . . . 13 ⊢ {𝐴} ∈ V
12	11	snid 4642	. . . . . . . . . . . 12 ⊢ {𝐴} ∈ {{𝐴}}
13	5, 12	eqeltri 2829	. . . . . . . . . . 11 ⊢ 𝐷 ∈ {{𝐴}}
14	3	symgid 19388	. . . . . . . . . . 11 ⊢ (𝐷 ∈ {{𝐴}} → ( I ↾ 𝐷) = (0_g‘𝐺))
15	13, 14	mp1i 13	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐷) = (0_g‘𝐺))
16		restidsing 6051	. . . . . . . . . . 11 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
17		xpsng 7139	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
18	17	anidms 566	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
19	16, 18	eqtrid 2781	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ {𝐴}) = {⟨𝐴, 𝐴⟩})
20	10, 15, 19	3eqtr3a 2793	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (0_g‘𝐺) = {⟨𝐴, 𝐴⟩})
21	20	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0_g‘𝐺) = {⟨𝐴, 𝐴⟩})
22		id 22	. . . . . . . . 9 ⊢ ({⟨𝐴, 𝐴⟩} = 𝑋 → {⟨𝐴, 𝐴⟩} = 𝑋)
23	22	eqcoms 2742	. . . . . . . 8 ⊢ (𝑋 = {⟨𝐴, 𝐴⟩} → {⟨𝐴, 𝐴⟩} = 𝑋)
24	21, 23	sylan9eqr 2791	. . . . . . 7 ⊢ ((𝑋 = {⟨𝐴, 𝐴⟩} ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵)) → (0_g‘𝐺) = 𝑋)
25	24	ex 412	. . . . . 6 ⊢ (𝑋 = {⟨𝐴, 𝐴⟩} → ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0_g‘𝐺) = 𝑋))
26	9, 25	syl 17	. . . . 5 ⊢ (𝑋 ∈ {{⟨𝐴, 𝐴⟩}} → ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0_g‘𝐺) = 𝑋))
27	8, 26	mpcom 38	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0_g‘𝐺) = 𝑋)
28	2, 27	eqtr2id 2782	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝐺 Σ_g ∅))
29	28	fveq2d 6890	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑁‘(𝐺 Σ_g ∅)))
30	5, 11	eqeltri 2829	. . . 4 ⊢ 𝐷 ∈ V
31		wrd0 14560	. . . 4 ⊢ ∅ ∈ Word ∅
32	30, 31	pm3.2i 470	. . 3 ⊢ (𝐷 ∈ V ∧ ∅ ∈ Word ∅)
33	5	fveq2i 6889	. . . . . . 7 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{𝐴})
34		pmtrsn 19506	. . . . . . 7 ⊢ (pmTrsp‘{𝐴}) = ∅
35	33, 34	eqtri 2757	. . . . . 6 ⊢ (pmTrsp‘𝐷) = ∅
36	35	rneqi 5928	. . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran ∅
37		rn0 5916	. . . . 5 ⊢ ran ∅ = ∅
38	36, 37	eqtr2i 2758	. . . 4 ⊢ ∅ = ran (pmTrsp‘𝐷)
39		psgnsn.n	. . . 4 ⊢ 𝑁 = (pmSgn‘𝐷)
40	3, 38, 39	psgnvalii 19496	. . 3 ⊢ ((𝐷 ∈ V ∧ ∅ ∈ Word ∅) → (𝑁‘(𝐺 Σ_g ∅)) = (-1↑(♯‘∅)))
41	32, 40	mp1i 13	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝐺 Σ_g ∅)) = (-1↑(♯‘∅)))
42		hash0 14389	. . . . 5 ⊢ (♯‘∅) = 0
43	42	oveq2i 7424	. . . 4 ⊢ (-1↑(♯‘∅)) = (-1↑0)
44		neg1cn 12362	. . . . 5 ⊢ -1 ∈ ℂ
45		exp0 14088	. . . . 5 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
46	44, 45	ax-mp 5	. . . 4 ⊢ (-1↑0) = 1
47	43, 46	eqtri 2757	. . 3 ⊢ (-1↑(♯‘∅)) = 1
48	47	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (-1↑(♯‘∅)) = 1)
49	29, 41, 48	3eqtrd 2773	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3463 ∅c0 4313 {csn 4606 ⟨cop 4612 I cid 5557 × cxp 5663 ran crn 5666 ↾ cres 5667 ‘cfv 6541 (class class class)co 7413 ℂcc 11135 0cc0 11137 1c1 11138 -cneg 11475 ↑cexp 14084 ♯chash 14352 Word cword 14535 Basecbs 17230 0_gc0g 17456 Σ_g cgsu 17457 SymGrpcsymg 19355 pmTrspcpmtr 19428 pmSgncpsgn 19476

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1511 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-ot 4615 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8727 df-map 8850 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-card 9961 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-div 11903 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12510 df-xnn0 12583 df-z 12597 df-uz 12861 df-rp 13017 df-fz 13530 df-fzo 13677 df-seq 14025 df-exp 14085 df-hash 14353 df-word 14536 df-lsw 14584 df-concat 14592 df-s1 14617 df-substr 14662 df-pfx 14692 df-splice 14771 df-reverse 14780 df-s2 14870 df-struct 17167 df-sets 17184 df-slot 17202 df-ndx 17214 df-base 17231 df-ress 17254 df-plusg 17287 df-tset 17293 df-0g 17458 df-gsum 17459 df-mre 17601 df-mrc 17602 df-acs 17604 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-efmnd 18852 df-grp 18924 df-minusg 18925 df-subg 19111 df-ghm 19201 df-gim 19247 df-oppg 19334 df-symg 19356 df-pmtr 19429 df-psgn 19478

This theorem is referenced by: m1detdiag 22552

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