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Theorem psgnsn 19746

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 2763	. . . . 5 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
2	1	gsum0 18885	. . . 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 19617	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐵 = {{⟨𝐴, 𝐴⟩}})
7	6	eleq2d 2849	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {{⟨𝐴, 𝐴⟩}}))
8	7	biimpa 482	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ {{⟨𝐴, 𝐴⟩}})
9		elsni 4600	. . . . . 6 ⊢ (𝑋 ∈ {{⟨𝐴, 𝐴⟩}} → 𝑋 = {⟨𝐴, 𝐴⟩})
10	5	reseq2i 5971	. . . . . . . . . 10 ⊢ ( I ↾ 𝐷) = ( I ↾ {𝐴})
11		snex 5400	. . . . . . . . . . . . 13 ⊢ {𝐴} ∈ V
12	11	snid 4622	. . . . . . . . . . . 12 ⊢ {𝐴} ∈ {{𝐴}}
13	5, 12	eqeltri 2859	. . . . . . . . . . 11 ⊢ 𝐷 ∈ {{𝐴}}
14	3	symgid 19627	. . . . . . . . . . 11 ⊢ (𝐷 ∈ {{𝐴}} → ( I ↾ 𝐷) = (0_g‘𝐺))
15	13, 14	mp1i 14	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐷) = (0_g‘𝐺))
16		restidsing 6050	. . . . . . . . . . 11 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
17		xpsng 7148	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
18	17	anidms 577	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
19	16, 18	eqtrid 2810	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ {𝐴}) = {⟨𝐴, 𝐴⟩})
20	10, 15, 19	3eqtr3a 2822	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (0_g‘𝐺) = {⟨𝐴, 𝐴⟩})
21	20	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0_g‘𝐺) = {⟨𝐴, 𝐴⟩})
22		id 23	. . . . . . . . 9 ⊢ ({⟨𝐴, 𝐴⟩} = 𝑋 → {⟨𝐴, 𝐴⟩} = 𝑋)
23	22	eqcoms 2771	. . . . . . . 8 ⊢ (𝑋 = {⟨𝐴, 𝐴⟩} → {⟨𝐴, 𝐴⟩} = 𝑋)
24	21, 23	sylan9eqr 2820	. . . . . . 7 ⊢ ((𝑋 = {⟨𝐴, 𝐴⟩} ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵)) → (0_g‘𝐺) = 𝑋)
25	24	ex 418	. . . . . 6 ⊢ (𝑋 = {⟨𝐴, 𝐴⟩} → ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0_g‘𝐺) = 𝑋))
26	9, 25	syl 18	. . . . 5 ⊢ (𝑋 ∈ {{⟨𝐴, 𝐴⟩}} → ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0_g‘𝐺) = 𝑋))
27	8, 26	mpcom 39	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0_g‘𝐺) = 𝑋)
28	2, 27	eqtr2id 2811	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝐺 Σ_g ∅))
29	28	fveq2d 6897	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑁‘(𝐺 Σ_g ∅)))
30	5, 11	eqeltri 2859	. . . 4 ⊢ 𝐷 ∈ V
31		wrd0 14694	. . . 4 ⊢ ∅ ∈ Word ∅
32	30, 31	pm3.2i 476	. . 3 ⊢ (𝐷 ∈ V ∧ ∅ ∈ Word ∅)
33	5	fveq2i 6896	. . . . . . 7 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{𝐴})
34		pmtrsn 19745	. . . . . . 7 ⊢ (pmTrsp‘{𝐴}) = ∅
35	33, 34	eqtri 2786	. . . . . 6 ⊢ (pmTrsp‘𝐷) = ∅
36	35	rneqi 5923	. . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran ∅
37		rn0 5912	. . . . 5 ⊢ ran ∅ = ∅
38	36, 37	eqtr2i 2787	. . . 4 ⊢ ∅ = ran (pmTrsp‘𝐷)
39		psgnsn.n	. . . 4 ⊢ 𝑁 = (pmSgn‘𝐷)
40	3, 38, 39	psgnvalii 19735	. . 3 ⊢ ((𝐷 ∈ V ∧ ∅ ∈ Word ∅) → (𝑁‘(𝐺 Σ_g ∅)) = ( -1↑(♯‘∅)))
41	32, 40	mp1i 14	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝐺 Σ_g ∅)) = ( -1↑(♯‘∅)))
42		hash0 14521	. . . . 5 ⊢ (♯‘∅) = 0
43	42	oveq2i 7438	. . . 4 ⊢ ( -1↑(♯‘∅)) = ( -1↑0)
44		neg1cn 12315	. . . . 5 ⊢ -1 ∈ ℂ
45		exp0 14218	. . . . 5 ⊢ ( -1 ∈ ℂ → ( -1↑0) = 1)
46	44, 45	ax-mp 5	. . . 4 ⊢ ( -1↑0) = 1
47	43, 46	eqtri 2786	. . 3 ⊢ ( -1↑(♯‘∅)) = 1
48	47	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → ( -1↑(♯‘∅)) = 1)
49	29, 41, 48	3eqtrd 2802	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2146 Vcvv 3453 ∅c0 4280 {csn 4587 ⟨cop 4837 I cid 5548 × cxp 5652 ran crn 5655 ↾ cres 5656 ‘cfv 6546 (class class class)co 7427 ℂcc 11208 0cc0 11210 1c1 11211 -cneg 11552 ↑cexp 14214 ♯chash 14484 Word cword 14668 Basecbs 17399 0_gc0g 17622 Σ_g cgsu 17623 SymGrpcsymg 19595 pmTrspcpmtr 19667 pmSgncpsgn 19715

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1541 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4838 df-ot 4861 df-uni 4870 df-int 4910 df-iun 4955 df-iin 4956 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-se 5608 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-1st 8008 df-2nd 8009 df-tpos 8254 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-er 8728 df-map 8860 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 10030 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-7 12420 df-8 12421 df-9 12422 df-n0 12617 df-xnn0 12690 df-z 12704 df-uz 12976 df-rp 13131 df-fz 13650 df-fzo 13799 df-seq 14155 df-exp 14215 df-hash 14485 df-word 14669 df-lsw 14718 df-concat 14726 df-s1 14753 df-substr 14800 df-pfx 14832 df-splice 14910 df-reverse 14919 df-s2 15010 df-struct 17337 df-sets 17354 df-slot 17372 df-ndx 17384 df-base 17400 df-ress 17421 df-plusg 17453 df-tset 17459 df-0g 17624 df-gsum 17625 df-mre 17768 df-mrc 17769 df-acs 17771 df-mgm 18828 df-sgrp 18920 df-mnd 18936 df-mhm 18990 df-submnd 18991 df-efmnd 19077 df-grp 19159 df-minusg 19160 df-subg 19345 df-ghm 19440 df-gim 19485 df-oppg 19572 df-symg 19596 df-pmtr 19668 df-psgn 19717

This theorem is used by: m1detdiag 22924

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