psgnsn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > psgnsn		Structured version Visualization version GIF version

Theorem psgnsn 19553

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 2735	. . . . 5 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
2	1	gsum0 18710	. . . 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 19423	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐵 = {{⟨𝐴, 𝐴⟩}})
7	6	eleq2d 2825	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {{⟨𝐴, 𝐴⟩}}))
8	7	biimpa 476	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ {{⟨𝐴, 𝐴⟩}})
9		elsni 4648	. . . . . 6 ⊢ (𝑋 ∈ {{⟨𝐴, 𝐴⟩}} → 𝑋 = {⟨𝐴, 𝐴⟩})
10	5	reseq2i 5997	. . . . . . . . . 10 ⊢ ( I ↾ 𝐷) = ( I ↾ {𝐴})
11		snex 5442	. . . . . . . . . . . . 13 ⊢ {𝐴} ∈ V
12	11	snid 4667	. . . . . . . . . . . 12 ⊢ {𝐴} ∈ {{𝐴}}
13	5, 12	eqeltri 2835	. . . . . . . . . . 11 ⊢ 𝐷 ∈ {{𝐴}}
14	3	symgid 19434	. . . . . . . . . . 11 ⊢ (𝐷 ∈ {{𝐴}} → ( I ↾ 𝐷) = (0_g‘𝐺))
15	13, 14	mp1i 13	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐷) = (0_g‘𝐺))
16		restidsing 6073	. . . . . . . . . . 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 2787	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ {𝐴}) = {⟨𝐴, 𝐴⟩})
20	10, 15, 19	3eqtr3a 2799	. . . . . . . . 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 2743	. . . . . . . 8 ⊢ (𝑋 = {⟨𝐴, 𝐴⟩} → {⟨𝐴, 𝐴⟩} = 𝑋)
24	21, 23	sylan9eqr 2797	. . . . . . 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 2788	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝐺 Σ_g ∅))
29	28	fveq2d 6911	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑁‘(𝐺 Σ_g ∅)))
30	5, 11	eqeltri 2835	. . . 4 ⊢ 𝐷 ∈ V
31		wrd0 14574	. . . 4 ⊢ ∅ ∈ Word ∅
32	30, 31	pm3.2i 470	. . 3 ⊢ (𝐷 ∈ V ∧ ∅ ∈ Word ∅)
33	5	fveq2i 6910	. . . . . . 7 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{𝐴})
34		pmtrsn 19552	. . . . . . 7 ⊢ (pmTrsp‘{𝐴}) = ∅
35	33, 34	eqtri 2763	. . . . . 6 ⊢ (pmTrsp‘𝐷) = ∅
36	35	rneqi 5951	. . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran ∅
37		rn0 5939	. . . . 5 ⊢ ran ∅ = ∅
38	36, 37	eqtr2i 2764	. . . 4 ⊢ ∅ = ran (pmTrsp‘𝐷)
39		psgnsn.n	. . . 4 ⊢ 𝑁 = (pmSgn‘𝐷)
40	3, 38, 39	psgnvalii 19542	. . 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 14403	. . . . 5 ⊢ (♯‘∅) = 0
43	42	oveq2i 7442	. . . 4 ⊢ (-1↑(♯‘∅)) = (-1↑0)
44		neg1cn 12378	. . . . 5 ⊢ -1 ∈ ℂ
45		exp0 14103	. . . . 5 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
46	44, 45	ax-mp 5	. . . 4 ⊢ (-1↑0) = 1
47	43, 46	eqtri 2763	. . 3 ⊢ (-1↑(♯‘∅)) = 1
48	47	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (-1↑(♯‘∅)) = 1)
49	29, 41, 48	3eqtrd 2779	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 ⟨cop 4637 I cid 5582 × cxp 5687 ran crn 5690 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 -cneg 11491 ↑cexp 14099 ♯chash 14366 Word cword 14549 Basecbs 17245 0_gc0g 17486 Σ_g cgsu 17487 SymGrpcsymg 19401 pmTrspcpmtr 19474 pmSgncpsgn 19522

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1509 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-word 14550 df-lsw 14598 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-splice 14785 df-reverse 14794 df-s2 14884 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-tset 17317 df-0g 17488 df-gsum 17489 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-efmnd 18895 df-grp 18967 df-minusg 18968 df-subg 19154 df-ghm 19244 df-gim 19290 df-oppg 19377 df-symg 19402 df-pmtr 19475 df-psgn 19524

This theorem is referenced by: m1detdiag 22619

W3C validator