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Theorem psgnsn 19538

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 2737	. . . . 5 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
2	1	gsum0 18697	. . . 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 19408	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐵 = {{⟨𝐴, 𝐴⟩}})
7	6	eleq2d 2827	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {{⟨𝐴, 𝐴⟩}}))
8	7	biimpa 476	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ {{⟨𝐴, 𝐴⟩}})
9		elsni 4643	. . . . . 6 ⊢ (𝑋 ∈ {{⟨𝐴, 𝐴⟩}} → 𝑋 = {⟨𝐴, 𝐴⟩})
10	5	reseq2i 5994	. . . . . . . . . 10 ⊢ ( I ↾ 𝐷) = ( I ↾ {𝐴})
11		snex 5436	. . . . . . . . . . . . 13 ⊢ {𝐴} ∈ V
12	11	snid 4662	. . . . . . . . . . . 12 ⊢ {𝐴} ∈ {{𝐴}}
13	5, 12	eqeltri 2837	. . . . . . . . . . 11 ⊢ 𝐷 ∈ {{𝐴}}
14	3	symgid 19419	. . . . . . . . . . 11 ⊢ (𝐷 ∈ {{𝐴}} → ( I ↾ 𝐷) = (0_g‘𝐺))
15	13, 14	mp1i 13	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐷) = (0_g‘𝐺))
16		restidsing 6071	. . . . . . . . . . 11 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
17		xpsng 7159	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
18	17	anidms 566	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
19	16, 18	eqtrid 2789	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ {𝐴}) = {⟨𝐴, 𝐴⟩})
20	10, 15, 19	3eqtr3a 2801	. . . . . . . . 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 2745	. . . . . . . 8 ⊢ (𝑋 = {⟨𝐴, 𝐴⟩} → {⟨𝐴, 𝐴⟩} = 𝑋)
24	21, 23	sylan9eqr 2799	. . . . . . 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 2790	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝐺 Σ_g ∅))
29	28	fveq2d 6910	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑁‘(𝐺 Σ_g ∅)))
30	5, 11	eqeltri 2837	. . . 4 ⊢ 𝐷 ∈ V
31		wrd0 14577	. . . 4 ⊢ ∅ ∈ Word ∅
32	30, 31	pm3.2i 470	. . 3 ⊢ (𝐷 ∈ V ∧ ∅ ∈ Word ∅)
33	5	fveq2i 6909	. . . . . . 7 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{𝐴})
34		pmtrsn 19537	. . . . . . 7 ⊢ (pmTrsp‘{𝐴}) = ∅
35	33, 34	eqtri 2765	. . . . . 6 ⊢ (pmTrsp‘𝐷) = ∅
36	35	rneqi 5948	. . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran ∅
37		rn0 5936	. . . . 5 ⊢ ran ∅ = ∅
38	36, 37	eqtr2i 2766	. . . 4 ⊢ ∅ = ran (pmTrsp‘𝐷)
39		psgnsn.n	. . . 4 ⊢ 𝑁 = (pmSgn‘𝐷)
40	3, 38, 39	psgnvalii 19527	. . 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 14406	. . . . 5 ⊢ (♯‘∅) = 0
43	42	oveq2i 7442	. . . 4 ⊢ (-1↑(♯‘∅)) = (-1↑0)
44		neg1cn 12380	. . . . 5 ⊢ -1 ∈ ℂ
45		exp0 14106	. . . . 5 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
46	44, 45	ax-mp 5	. . . 4 ⊢ (-1↑0) = 1
47	43, 46	eqtri 2765	. . 3 ⊢ (-1↑(♯‘∅)) = 1
48	47	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (-1↑(♯‘∅)) = 1)
49	29, 41, 48	3eqtrd 2781	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 ⟨cop 4632 I cid 5577 × cxp 5683 ran crn 5686 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 -cneg 11493 ↑cexp 14102 ♯chash 14369 Word cword 14552 Basecbs 17247 0_gc0g 17484 Σ_g cgsu 17485 SymGrpcsymg 19386 pmTrspcpmtr 19459 pmSgncpsgn 19507

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-splice 14788 df-reverse 14797 df-s2 14887 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-tset 17316 df-0g 17486 df-gsum 17487 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-efmnd 18882 df-grp 18954 df-minusg 18955 df-subg 19141 df-ghm 19231 df-gim 19277 df-oppg 19364 df-symg 19387 df-pmtr 19460 df-psgn 19509

This theorem is referenced by: m1detdiag 22603

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