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 19128

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 2738	. . . . 5 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
2	1	gsum0 18368	. . . 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 18998	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐵 = {{⟨𝐴, 𝐴⟩}})
7	6	eleq2d 2824	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {{⟨𝐴, 𝐴⟩}}))
8	7	biimpa 477	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ {{⟨𝐴, 𝐴⟩}})
9		elsni 4578	. . . . . 6 ⊢ (𝑋 ∈ {{⟨𝐴, 𝐴⟩}} → 𝑋 = {⟨𝐴, 𝐴⟩})
10	5	reseq2i 5888	. . . . . . . . . 10 ⊢ ( I ↾ 𝐷) = ( I ↾ {𝐴})
11		snex 5354	. . . . . . . . . . . . 13 ⊢ {𝐴} ∈ V
12	11	snid 4597	. . . . . . . . . . . 12 ⊢ {𝐴} ∈ {{𝐴}}
13	5, 12	eqeltri 2835	. . . . . . . . . . 11 ⊢ 𝐷 ∈ {{𝐴}}
14	3	symgid 19009	. . . . . . . . . . 11 ⊢ (𝐷 ∈ {{𝐴}} → ( I ↾ 𝐷) = (0_g‘𝐺))
15	13, 14	mp1i 13	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐷) = (0_g‘𝐺))
16		restidsing 5962	. . . . . . . . . . 11 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
17		xpsng 7011	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
18	17	anidms 567	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
19	16, 18	eqtrid 2790	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ {𝐴}) = {⟨𝐴, 𝐴⟩})
20	10, 15, 19	3eqtr3a 2802	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (0_g‘𝐺) = {⟨𝐴, 𝐴⟩})
21	20	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0_g‘𝐺) = {⟨𝐴, 𝐴⟩})
22		id 22	. . . . . . . . 9 ⊢ ({⟨𝐴, 𝐴⟩} = 𝑋 → {⟨𝐴, 𝐴⟩} = 𝑋)
23	22	eqcoms 2746	. . . . . . . 8 ⊢ (𝑋 = {⟨𝐴, 𝐴⟩} → {⟨𝐴, 𝐴⟩} = 𝑋)
24	21, 23	sylan9eqr 2800	. . . . . . 7 ⊢ ((𝑋 = {⟨𝐴, 𝐴⟩} ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵)) → (0_g‘𝐺) = 𝑋)
25	24	ex 413	. . . . . 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 2791	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝐺 Σ_g ∅))
29	28	fveq2d 6778	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑁‘(𝐺 Σ_g ∅)))
30	5, 11	eqeltri 2835	. . . 4 ⊢ 𝐷 ∈ V
31		wrd0 14242	. . . 4 ⊢ ∅ ∈ Word ∅
32	30, 31	pm3.2i 471	. . 3 ⊢ (𝐷 ∈ V ∧ ∅ ∈ Word ∅)
33	5	fveq2i 6777	. . . . . . 7 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{𝐴})
34		pmtrsn 19127	. . . . . . 7 ⊢ (pmTrsp‘{𝐴}) = ∅
35	33, 34	eqtri 2766	. . . . . 6 ⊢ (pmTrsp‘𝐷) = ∅
36	35	rneqi 5846	. . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran ∅
37		rn0 5835	. . . . 5 ⊢ ran ∅ = ∅
38	36, 37	eqtr2i 2767	. . . 4 ⊢ ∅ = ran (pmTrsp‘𝐷)
39		psgnsn.n	. . . 4 ⊢ 𝑁 = (pmSgn‘𝐷)
40	3, 38, 39	psgnvalii 19117	. . 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 14082	. . . . 5 ⊢ (♯‘∅) = 0
43	42	oveq2i 7286	. . . 4 ⊢ (-1↑(♯‘∅)) = (-1↑0)
44		neg1cn 12087	. . . . 5 ⊢ -1 ∈ ℂ
45		exp0 13786	. . . . 5 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
46	44, 45	ax-mp 5	. . . 4 ⊢ (-1↑0) = 1
47	43, 46	eqtri 2766	. . 3 ⊢ (-1↑(♯‘∅)) = 1
48	47	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (-1↑(♯‘∅)) = 1)
49	29, 41, 48	3eqtrd 2782	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∅c0 4256 {csn 4561 ⟨cop 4567 I cid 5488 × cxp 5587 ran crn 5590 ↾ cres 5591 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 0cc0 10871 1c1 10872 -cneg 11206 ↑cexp 13782 ♯chash 14044 Word cword 14217 Basecbs 16912 0_gc0g 17150 Σ_g cgsu 17151 SymGrpcsymg 18974 pmTrspcpmtr 19049 pmSgncpsgn 19097

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-xor 1507 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-word 14218 df-lsw 14266 df-concat 14274 df-s1 14301 df-substr 14354 df-pfx 14384 df-splice 14463 df-reverse 14472 df-s2 14561 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-tset 16981 df-0g 17152 df-gsum 17153 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-efmnd 18508 df-grp 18580 df-minusg 18581 df-subg 18752 df-ghm 18832 df-gim 18875 df-oppg 18950 df-symg 18975 df-pmtr 19050 df-psgn 19099

This theorem is referenced by: m1detdiag 21746

W3C validator