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Mirrors > Home > MPE Home > Th. List > psgn0fv0 | Structured version Visualization version GIF version |
Description: The permutation sign function for an empty set at an empty set is 1. (Contributed by AV, 27-Feb-2019.) |
Ref | Expression |
---|---|
psgn0fv0 | ⊢ ((pmSgn‘∅)‘∅) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5313 | . 2 ⊢ ∅ ∈ V | |
2 | wrd0 14574 | . 2 ⊢ ∅ ∈ Word ran (pmTrsp‘∅) | |
3 | eqid 2735 | . . . . . 6 ⊢ (0g‘(SymGrp‘∅)) = (0g‘(SymGrp‘∅)) | |
4 | 3 | gsum0 18710 | . . . . 5 ⊢ ((SymGrp‘∅) Σg ∅) = (0g‘(SymGrp‘∅)) |
5 | eqid 2735 | . . . . . . . . 9 ⊢ (SymGrp‘∅) = (SymGrp‘∅) | |
6 | 5 | symgid 19434 | . . . . . . . 8 ⊢ (∅ ∈ V → ( I ↾ ∅) = (0g‘(SymGrp‘∅))) |
7 | 1, 6 | ax-mp 5 | . . . . . . 7 ⊢ ( I ↾ ∅) = (0g‘(SymGrp‘∅)) |
8 | res0 6004 | . . . . . . 7 ⊢ ( I ↾ ∅) = ∅ | |
9 | 7, 8 | eqtr3i 2765 | . . . . . 6 ⊢ (0g‘(SymGrp‘∅)) = ∅ |
10 | 9 | a1i 11 | . . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → (0g‘(SymGrp‘∅)) = ∅) |
11 | 4, 10 | eqtr2id 2788 | . . . 4 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ∅ = ((SymGrp‘∅) Σg ∅)) |
12 | 11 | fveq2d 6911 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ((pmSgn‘∅)‘∅) = ((pmSgn‘∅)‘((SymGrp‘∅) Σg ∅))) |
13 | eqid 2735 | . . . 4 ⊢ ran (pmTrsp‘∅) = ran (pmTrsp‘∅) | |
14 | eqid 2735 | . . . 4 ⊢ (pmSgn‘∅) = (pmSgn‘∅) | |
15 | 5, 13, 14 | psgnvalii 19542 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ((pmSgn‘∅)‘((SymGrp‘∅) Σg ∅)) = (-1↑(♯‘∅))) |
16 | hash0 14403 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
17 | 16 | oveq2i 7442 | . . . . 5 ⊢ (-1↑(♯‘∅)) = (-1↑0) |
18 | neg1cn 12378 | . . . . . 6 ⊢ -1 ∈ ℂ | |
19 | exp0 14103 | . . . . . 6 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
20 | 18, 19 | ax-mp 5 | . . . . 5 ⊢ (-1↑0) = 1 |
21 | 17, 20 | eqtri 2763 | . . . 4 ⊢ (-1↑(♯‘∅)) = 1 |
22 | 21 | a1i 11 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → (-1↑(♯‘∅)) = 1) |
23 | 12, 15, 22 | 3eqtrd 2779 | . 2 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ((pmSgn‘∅)‘∅) = 1) |
24 | 1, 2, 23 | mp2an 692 | 1 ⊢ ((pmSgn‘∅)‘∅) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 I cid 5582 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 0gc0g 17486 Σg 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: mdet0pr 22614 |
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