| 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 2761 | . . . 4 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷) | |
| 2 | 1 | symgid 19431 | . . 3 ⊢ (𝐷 ∈ Fin → ( I ↾ 𝐷) = (0g‘(SymGrp‘𝐷))) |
| 3 | 2 | fveq2d 6865 | . 2 ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = (𝑆‘(0g‘(SymGrp‘𝐷)))) |
| 4 | psgnid.s | . . . 4 ⊢ 𝑆 = (pmSgn‘𝐷) | |
| 5 | eqid 2761 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
| 6 | 1, 4, 5 | psgnghm2 21620 | . . 3 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ ((SymGrp‘𝐷) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
| 7 | eqid 2761 | . . . 4 ⊢ (0g‘(SymGrp‘𝐷)) = (0g‘(SymGrp‘𝐷)) | |
| 8 | cnring 21433 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
| 9 | eqid 2761 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
| 10 | 9 | ringmgp 20275 | . . . . . 6 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
| 11 | 8, 10 | ax-mp 5 | . . . . 5 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
| 12 | 1ex 11169 | . . . . . 6 ⊢ 1 ∈ V | |
| 13 | 12 | prid1 4718 | . . . . 5 ⊢ 1 ∈ {1, -1} |
| 14 | ax-1cn 11124 | . . . . . 6 ⊢ 1 ∈ ℂ | |
| 15 | neg1cn 12173 | . . . . . 6 ⊢ -1 ∈ ℂ | |
| 16 | prssi 4776 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ) → {1, -1} ⊆ ℂ) | |
| 17 | 14, 15, 16 | mp2an 702 | . . . . 5 ⊢ {1, -1} ⊆ ℂ |
| 18 | cnfldbas 21415 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
| 19 | 9, 18 | mgpbas 20181 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
| 20 | cnfld1 21436 | . . . . . . 7 ⊢ 1 = (1r‘ℂfld) | |
| 21 | 9, 20 | ringidval 20219 | . . . . . 6 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
| 22 | 5, 19, 21 | ress0g 18786 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ 1 ∈ {1, -1} ∧ {1, -1} ⊆ ℂ) → 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
| 23 | 11, 13, 17, 22 | mp3an 1481 | . . . 4 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1})) |
| 24 | 7, 23 | ghmid 19252 | . . 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 2796 | 1 ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 ⊆ wss 3902 {cpr 4581 I cid 5537 ↾ cres 5645 ‘cfv 6515 (class class class)co 7390 Fincfn 8920 ℂcc 11064 1c1 11067 -cneg 11408 ↾s cress 17256 0gc0g 17458 Mndcmnd 18758 GrpHom cghm 19243 SymGrpcsymg 19399 pmSgncpsgn 19519 mulGrpcmgp 20176 Ringcrg 20269 ℂfldccnfld 21411 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-addf 11145 ax-mulf 11146 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-xor 1531 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-ot 4588 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-1st 7964 df-2nd 7965 df-tpos 8199 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-xnn0 12548 df-z 12562 df-dec 12682 df-uz 12833 df-rp 12987 df-fz 13506 df-fzo 13653 df-seq 14008 df-exp 14068 df-hash 14337 df-word 14520 df-lsw 14569 df-concat 14577 df-s1 14603 df-substr 14648 df-pfx 14678 df-splice 14756 df-reverse 14765 df-s2 14854 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-0g 17460 df-gsum 17461 df-mre 17604 df-mrc 17605 df-acs 17607 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-mhm 18807 df-submnd 18808 df-efmnd 18893 df-grp 18968 df-minusg 18969 df-subg 19155 df-ghm 19244 df-gim 19289 df-oppg 19376 df-symg 19400 df-pmtr 19472 df-psgn 19521 df-cmn 19812 df-abl 19813 df-mgp 20177 df-rng 20189 df-ur 20218 df-ring 20271 df-cring 20272 df-oppr 20372 df-dvdsr 20392 df-unit 20393 df-invr 20423 df-dvr 20436 df-drng 20767 df-cnfld 21412 |
| This theorem is referenced by: psgnfzto1st 33245 evpmid 33288 |
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