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Mirrors > Home > MPE Home > Th. List > psgnsn | Structured version Visualization version GIF version |
Description: The permutation sign function for a singleton. (Contributed by AV, 6-Aug-2019.) |
Ref | Expression |
---|---|
psgnsn.0 | ⊢ 𝐷 = {𝐴} |
psgnsn.g | ⊢ 𝐺 = (SymGrp‘𝐷) |
psgnsn.b | ⊢ 𝐵 = (Base‘𝐺) |
psgnsn.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnsn | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
2 | 1 | gsum0 18615 | . . . 4 ⊢ (𝐺 Σg ∅) = (0g‘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 19306 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐵 = {{〈𝐴, 𝐴〉}}) |
7 | 6 | eleq2d 2818 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {{〈𝐴, 𝐴〉}})) |
8 | 7 | biimpa 476 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ {{〈𝐴, 𝐴〉}}) |
9 | elsni 4645 | . . . . . 6 ⊢ (𝑋 ∈ {{〈𝐴, 𝐴〉}} → 𝑋 = {〈𝐴, 𝐴〉}) | |
10 | 5 | reseq2i 5978 | . . . . . . . . . 10 ⊢ ( I ↾ 𝐷) = ( I ↾ {𝐴}) |
11 | snex 5431 | . . . . . . . . . . . . 13 ⊢ {𝐴} ∈ V | |
12 | 11 | snid 4664 | . . . . . . . . . . . 12 ⊢ {𝐴} ∈ {{𝐴}} |
13 | 5, 12 | eqeltri 2828 | . . . . . . . . . . 11 ⊢ 𝐷 ∈ {{𝐴}} |
14 | 3 | symgid 19317 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ {{𝐴}} → ( I ↾ 𝐷) = (0g‘𝐺)) |
15 | 13, 14 | mp1i 13 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐷) = (0g‘𝐺)) |
16 | restidsing 6052 | . . . . . . . . . . 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 2783 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ {𝐴}) = {〈𝐴, 𝐴〉}) |
20 | 10, 15, 19 | 3eqtr3a 2795 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (0g‘𝐺) = {〈𝐴, 𝐴〉}) |
21 | 20 | adantr 480 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0g‘𝐺) = {〈𝐴, 𝐴〉}) |
22 | id 22 | . . . . . . . . 9 ⊢ ({〈𝐴, 𝐴〉} = 𝑋 → {〈𝐴, 𝐴〉} = 𝑋) | |
23 | 22 | eqcoms 2739 | . . . . . . . 8 ⊢ (𝑋 = {〈𝐴, 𝐴〉} → {〈𝐴, 𝐴〉} = 𝑋) |
24 | 21, 23 | sylan9eqr 2793 | . . . . . . 7 ⊢ ((𝑋 = {〈𝐴, 𝐴〉} ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵)) → (0g‘𝐺) = 𝑋) |
25 | 24 | ex 412 | . . . . . 6 ⊢ (𝑋 = {〈𝐴, 𝐴〉} → ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0g‘𝐺) = 𝑋)) |
26 | 9, 25 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ {{〈𝐴, 𝐴〉}} → ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0g‘𝐺) = 𝑋)) |
27 | 8, 26 | mpcom 38 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0g‘𝐺) = 𝑋) |
28 | 2, 27 | eqtr2id 2784 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝐺 Σg ∅)) |
29 | 28 | fveq2d 6895 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑁‘(𝐺 Σg ∅))) |
30 | 5, 11 | eqeltri 2828 | . . . 4 ⊢ 𝐷 ∈ V |
31 | wrd0 14496 | . . . 4 ⊢ ∅ ∈ Word ∅ | |
32 | 30, 31 | pm3.2i 470 | . . 3 ⊢ (𝐷 ∈ V ∧ ∅ ∈ Word ∅) |
33 | 5 | fveq2i 6894 | . . . . . . 7 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{𝐴}) |
34 | pmtrsn 19435 | . . . . . . 7 ⊢ (pmTrsp‘{𝐴}) = ∅ | |
35 | 33, 34 | eqtri 2759 | . . . . . 6 ⊢ (pmTrsp‘𝐷) = ∅ |
36 | 35 | rneqi 5936 | . . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran ∅ |
37 | rn0 5925 | . . . . 5 ⊢ ran ∅ = ∅ | |
38 | 36, 37 | eqtr2i 2760 | . . . 4 ⊢ ∅ = ran (pmTrsp‘𝐷) |
39 | psgnsn.n | . . . 4 ⊢ 𝑁 = (pmSgn‘𝐷) | |
40 | 3, 38, 39 | psgnvalii 19425 | . . 3 ⊢ ((𝐷 ∈ V ∧ ∅ ∈ Word ∅) → (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅))) |
41 | 32, 40 | mp1i 13 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅))) |
42 | hash0 14334 | . . . . 5 ⊢ (♯‘∅) = 0 | |
43 | 42 | oveq2i 7423 | . . . 4 ⊢ (-1↑(♯‘∅)) = (-1↑0) |
44 | neg1cn 12333 | . . . . 5 ⊢ -1 ∈ ℂ | |
45 | exp0 14038 | . . . . 5 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
46 | 44, 45 | ax-mp 5 | . . . 4 ⊢ (-1↑0) = 1 |
47 | 43, 46 | eqtri 2759 | . . 3 ⊢ (-1↑(♯‘∅)) = 1 |
48 | 47 | a1i 11 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (-1↑(♯‘∅)) = 1) |
49 | 29, 41, 48 | 3eqtrd 2775 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ∅c0 4322 {csn 4628 〈cop 4634 I cid 5573 × cxp 5674 ran crn 5677 ↾ cres 5678 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 0cc0 11116 1c1 11117 -cneg 11452 ↑cexp 14034 ♯chash 14297 Word cword 14471 Basecbs 17151 0gc0g 17392 Σg cgsu 17393 SymGrpcsymg 19282 pmTrspcpmtr 19357 pmSgncpsgn 19405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-xor 1509 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-xnn0 12552 df-z 12566 df-uz 12830 df-rp 12982 df-fz 13492 df-fzo 13635 df-seq 13974 df-exp 14035 df-hash 14298 df-word 14472 df-lsw 14520 df-concat 14528 df-s1 14553 df-substr 14598 df-pfx 14628 df-splice 14707 df-reverse 14716 df-s2 14806 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-tset 17223 df-0g 17394 df-gsum 17395 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-efmnd 18792 df-grp 18864 df-minusg 18865 df-subg 19046 df-ghm 19135 df-gim 19180 df-oppg 19258 df-symg 19283 df-pmtr 19358 df-psgn 19407 |
This theorem is referenced by: m1detdiag 22419 |
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