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Mirrors > Home > MPE Home > Th. List > psgnprfval1 | Structured version Visualization version GIF version |
Description: The permutation sign of the identity for a pair. (Contributed by AV, 11-Dec-2018.) |
Ref | Expression |
---|---|
psgnprfval.0 | ⊢ 𝐷 = {1, 2} |
psgnprfval.g | ⊢ 𝐺 = (SymGrp‘𝐷) |
psgnprfval.b | ⊢ 𝐵 = (Base‘𝐺) |
psgnprfval.t | ⊢ 𝑇 = ran (pmTrsp‘𝐷) |
psgnprfval.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnprfval1 | ⊢ (𝑁‘{〈1, 1〉, 〈2, 2〉}) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psgnprfval.0 | . . . . . . 7 ⊢ 𝐷 = {1, 2} | |
2 | prex 5434 | . . . . . . 7 ⊢ {1, 2} ∈ V | |
3 | 1, 2 | eqeltri 2821 | . . . . . 6 ⊢ 𝐷 ∈ V |
4 | psgnprfval.g | . . . . . . 7 ⊢ 𝐺 = (SymGrp‘𝐷) | |
5 | 4 | symgid 19368 | . . . . . 6 ⊢ (𝐷 ∈ V → ( I ↾ 𝐷) = (0g‘𝐺)) |
6 | 3, 5 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ 𝐷) = (0g‘𝐺) |
7 | 6 | gsum0 18647 | . . . 4 ⊢ (𝐺 Σg ∅) = ( I ↾ 𝐷) |
8 | reseq2 5980 | . . . . . 6 ⊢ (𝐷 = {1, 2} → ( I ↾ 𝐷) = ( I ↾ {1, 2})) | |
9 | 1ex 11242 | . . . . . . 7 ⊢ 1 ∈ V | |
10 | 2nn 12318 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
11 | residpr 7152 | . . . . . . 7 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → ( I ↾ {1, 2}) = {〈1, 1〉, 〈2, 2〉}) | |
12 | 9, 10, 11 | mp2an 690 | . . . . . 6 ⊢ ( I ↾ {1, 2}) = {〈1, 1〉, 〈2, 2〉} |
13 | 8, 12 | eqtrdi 2781 | . . . . 5 ⊢ (𝐷 = {1, 2} → ( I ↾ 𝐷) = {〈1, 1〉, 〈2, 2〉}) |
14 | 1, 13 | ax-mp 5 | . . . 4 ⊢ ( I ↾ 𝐷) = {〈1, 1〉, 〈2, 2〉} |
15 | 7, 14 | eqtr2i 2754 | . . 3 ⊢ {〈1, 1〉, 〈2, 2〉} = (𝐺 Σg ∅) |
16 | 15 | fveq2i 6899 | . 2 ⊢ (𝑁‘{〈1, 1〉, 〈2, 2〉}) = (𝑁‘(𝐺 Σg ∅)) |
17 | wrd0 14525 | . . 3 ⊢ ∅ ∈ Word 𝑇 | |
18 | psgnprfval.t | . . . 4 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
19 | psgnprfval.n | . . . 4 ⊢ 𝑁 = (pmSgn‘𝐷) | |
20 | 4, 18, 19 | psgnvalii 19476 | . . 3 ⊢ ((𝐷 ∈ V ∧ ∅ ∈ Word 𝑇) → (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅))) |
21 | 3, 17, 20 | mp2an 690 | . 2 ⊢ (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅)) |
22 | hash0 14362 | . . . 4 ⊢ (♯‘∅) = 0 | |
23 | 22 | oveq2i 7430 | . . 3 ⊢ (-1↑(♯‘∅)) = (-1↑0) |
24 | neg1cn 12359 | . . . 4 ⊢ -1 ∈ ℂ | |
25 | exp0 14066 | . . . 4 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
26 | 24, 25 | ax-mp 5 | . . 3 ⊢ (-1↑0) = 1 |
27 | 23, 26 | eqtri 2753 | . 2 ⊢ (-1↑(♯‘∅)) = 1 |
28 | 16, 21, 27 | 3eqtri 2757 | 1 ⊢ (𝑁‘{〈1, 1〉, 〈2, 2〉}) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∅c0 4322 {cpr 4632 〈cop 4636 I cid 5575 ran crn 5679 ↾ cres 5680 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 -cneg 11477 ℕcn 12245 2c2 12300 ↑cexp 14062 ♯chash 14325 Word cword 14500 Basecbs 17183 0gc0g 17424 Σg cgsu 17425 SymGrpcsymg 19333 pmTrspcpmtr 19408 pmSgncpsgn 19456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-xnn0 12578 df-z 12592 df-uz 12856 df-rp 13010 df-fz 13520 df-fzo 13663 df-seq 14003 df-exp 14063 df-hash 14326 df-word 14501 df-lsw 14549 df-concat 14557 df-s1 14582 df-substr 14627 df-pfx 14657 df-splice 14736 df-reverse 14745 df-s2 14835 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-tset 17255 df-0g 17426 df-gsum 17427 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-submnd 18744 df-efmnd 18829 df-grp 18901 df-minusg 18902 df-subg 19086 df-ghm 19176 df-gim 19222 df-oppg 19309 df-symg 19334 df-pmtr 19409 df-psgn 19458 |
This theorem is referenced by: m2detleiblem1 22570 m2detleiblem5 22571 |
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