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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 5217 | . . . . . . 7 ⊢ {1, 2} ∈ V | |
3 | 1, 2 | eqeltri 2877 | . . . . . 6 ⊢ 𝐷 ∈ V |
4 | psgnprfval.g | . . . . . . 7 ⊢ 𝐺 = (SymGrp‘𝐷) | |
5 | 4 | symgid 18248 | . . . . . 6 ⊢ (𝐷 ∈ V → ( I ↾ 𝐷) = (0g‘𝐺)) |
6 | 3, 5 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ 𝐷) = (0g‘𝐺) |
7 | 6 | gsum0 17705 | . . . 4 ⊢ (𝐺 Σg ∅) = ( I ↾ 𝐷) |
8 | reseq2 5721 | . . . . . 6 ⊢ (𝐷 = {1, 2} → ( I ↾ 𝐷) = ( I ↾ {1, 2})) | |
9 | 1ex 10472 | . . . . . . 7 ⊢ 1 ∈ V | |
10 | 2nn 11547 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
11 | residpr 6759 | . . . . . . 7 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → ( I ↾ {1, 2}) = {〈1, 1〉, 〈2, 2〉}) | |
12 | 9, 10, 11 | mp2an 688 | . . . . . 6 ⊢ ( I ↾ {1, 2}) = {〈1, 1〉, 〈2, 2〉} |
13 | 8, 12 | syl6eq 2845 | . . . . 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 2818 | . . 3 ⊢ {〈1, 1〉, 〈2, 2〉} = (𝐺 Σg ∅) |
16 | 15 | fveq2i 6533 | . 2 ⊢ (𝑁‘{〈1, 1〉, 〈2, 2〉}) = (𝑁‘(𝐺 Σg ∅)) |
17 | wrd0 13723 | . . 3 ⊢ ∅ ∈ Word 𝑇 | |
18 | psgnprfval.t | . . . 4 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
19 | psgnprfval.n | . . . 4 ⊢ 𝑁 = (pmSgn‘𝐷) | |
20 | 4, 18, 19 | psgnvalii 18356 | . . 3 ⊢ ((𝐷 ∈ V ∧ ∅ ∈ Word 𝑇) → (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅))) |
21 | 3, 17, 20 | mp2an 688 | . 2 ⊢ (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅)) |
22 | hash0 13566 | . . . 4 ⊢ (♯‘∅) = 0 | |
23 | 22 | oveq2i 7018 | . . 3 ⊢ (-1↑(♯‘∅)) = (-1↑0) |
24 | neg1cn 11588 | . . . 4 ⊢ -1 ∈ ℂ | |
25 | exp0 13271 | . . . 4 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
26 | 24, 25 | ax-mp 5 | . . 3 ⊢ (-1↑0) = 1 |
27 | 23, 26 | eqtri 2817 | . 2 ⊢ (-1↑(♯‘∅)) = 1 |
28 | 16, 21, 27 | 3eqtri 2821 | 1 ⊢ (𝑁‘{〈1, 1〉, 〈2, 2〉}) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1520 ∈ wcel 2079 Vcvv 3432 ∅c0 4206 {cpr 4468 〈cop 4472 I cid 5339 ran crn 5436 ↾ cres 5437 ‘cfv 6217 (class class class)co 7007 ℂcc 10370 0cc0 10372 1c1 10373 -cneg 10707 ℕcn 11475 2c2 11529 ↑cexp 13267 ♯chash 13528 Word cword 13695 Basecbs 16300 0gc0g 16530 Σg cgsu 16531 SymGrpcsymg 18224 pmTrspcpmtr 18288 pmSgncpsgn 18336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-xor 1495 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-ot 4475 df-uni 4740 df-int 4777 df-iun 4821 df-iin 4822 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-tpos 7734 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-2o 7945 df-oadd 7948 df-er 8130 df-map 8249 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-xnn0 11805 df-z 11819 df-uz 12083 df-rp 12229 df-fz 12732 df-fzo 12873 df-seq 13208 df-exp 13268 df-hash 13529 df-word 13696 df-lsw 13749 df-concat 13757 df-s1 13782 df-substr 13827 df-pfx 13857 df-splice 13936 df-reverse 13945 df-s2 14034 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-tset 16401 df-0g 16532 df-gsum 16533 df-mre 16674 df-mrc 16675 df-acs 16677 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-mhm 17762 df-submnd 17763 df-grp 17852 df-minusg 17853 df-subg 18018 df-ghm 18085 df-gim 18128 df-oppg 18203 df-symg 18225 df-pmtr 18289 df-psgn 18338 |
This theorem is referenced by: m2detleiblem1 20905 m2detleiblem5 20906 |
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