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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > psgnid | Structured version Visualization version GIF version |
Description: Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
Ref | Expression |
---|---|
psgnid.s | ⊢ 𝑆 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnid | ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . 4 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷) | |
2 | 1 | symgid 19434 | . . 3 ⊢ (𝐷 ∈ Fin → ( I ↾ 𝐷) = (0g‘(SymGrp‘𝐷))) |
3 | 2 | fveq2d 6911 | . 2 ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = (𝑆‘(0g‘(SymGrp‘𝐷)))) |
4 | psgnid.s | . . . 4 ⊢ 𝑆 = (pmSgn‘𝐷) | |
5 | eqid 2735 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
6 | 1, 4, 5 | psgnghm2 21617 | . . 3 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ ((SymGrp‘𝐷) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
7 | eqid 2735 | . . . 4 ⊢ (0g‘(SymGrp‘𝐷)) = (0g‘(SymGrp‘𝐷)) | |
8 | cnring 21421 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
9 | eqid 2735 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
10 | 9 | ringmgp 20257 | . . . . . 6 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
11 | 8, 10 | ax-mp 5 | . . . . 5 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
12 | 1ex 11255 | . . . . . 6 ⊢ 1 ∈ V | |
13 | 12 | prid1 4767 | . . . . 5 ⊢ 1 ∈ {1, -1} |
14 | ax-1cn 11211 | . . . . . 6 ⊢ 1 ∈ ℂ | |
15 | neg1cn 12378 | . . . . . 6 ⊢ -1 ∈ ℂ | |
16 | prssi 4826 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ) → {1, -1} ⊆ ℂ) | |
17 | 14, 15, 16 | mp2an 692 | . . . . 5 ⊢ {1, -1} ⊆ ℂ |
18 | cnfldbas 21386 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
19 | 9, 18 | mgpbas 20158 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
20 | cnfld1 21424 | . . . . . . 7 ⊢ 1 = (1r‘ℂfld) | |
21 | 9, 20 | ringidval 20201 | . . . . . 6 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
22 | 5, 19, 21 | ress0g 18788 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ 1 ∈ {1, -1} ∧ {1, -1} ⊆ ℂ) → 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
23 | 11, 13, 17, 22 | mp3an 1460 | . . . 4 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1})) |
24 | 7, 23 | ghmid 19253 | . . 3 ⊢ (𝑆 ∈ ((SymGrp‘𝐷) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → (𝑆‘(0g‘(SymGrp‘𝐷))) = 1) |
25 | 6, 24 | syl 17 | . 2 ⊢ (𝐷 ∈ Fin → (𝑆‘(0g‘(SymGrp‘𝐷))) = 1) |
26 | 3, 25 | eqtrd 2775 | 1 ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 {cpr 4633 I cid 5582 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℂcc 11151 1c1 11154 -cneg 11491 ↾s cress 17274 0gc0g 17486 Mndcmnd 18760 GrpHom cghm 19243 SymGrpcsymg 19401 pmSgncpsgn 19522 mulGrpcmgp 20152 Ringcrg 20251 ℂfldccnfld 21382 |
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 ax-addf 11232 ax-mulf 11233 |
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-dec 12732 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-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 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 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-drng 20748 df-cnfld 21383 |
This theorem is referenced by: psgnfzto1st 33108 evpmid 33151 |
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