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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > psgndmfi | Structured version Visualization version GIF version |
Description: For a finite base set, the permutation sign is defined for all permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
Ref | Expression |
---|---|
psgndmfi.s | ⊢ 𝑆 = (pmSgn‘𝐷) |
psgndmfi.g | ⊢ 𝐺 = (Base‘(SymGrp‘𝐷)) |
Ref | Expression |
---|---|
psgndmfi | ⊢ (𝐷 ∈ Fin → 𝑆 Fn 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2795 | . . 3 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷) | |
2 | psgndmfi.g | . . 3 ⊢ 𝐺 = (Base‘(SymGrp‘𝐷)) | |
3 | eqid 2795 | . . 3 ⊢ {𝑝 ∈ 𝐺 ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝 ∈ 𝐺 ∣ dom (𝑝 ∖ I ) ∈ Fin} | |
4 | psgndmfi.s | . . 3 ⊢ 𝑆 = (pmSgn‘𝐷) | |
5 | 1, 2, 3, 4 | psgnfn 18360 | . 2 ⊢ 𝑆 Fn {𝑝 ∈ 𝐺 ∣ dom (𝑝 ∖ I ) ∈ Fin} |
6 | 1, 2 | sygbasnfpfi 18371 | . . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑝 ∈ 𝐺) → dom (𝑝 ∖ I ) ∈ Fin) |
7 | 6 | ralrimiva 3149 | . . . . 5 ⊢ (𝐷 ∈ Fin → ∀𝑝 ∈ 𝐺 dom (𝑝 ∖ I ) ∈ Fin) |
8 | rabid2 3340 | . . . . 5 ⊢ (𝐺 = {𝑝 ∈ 𝐺 ∣ dom (𝑝 ∖ I ) ∈ Fin} ↔ ∀𝑝 ∈ 𝐺 dom (𝑝 ∖ I ) ∈ Fin) | |
9 | 7, 8 | sylibr 235 | . . . 4 ⊢ (𝐷 ∈ Fin → 𝐺 = {𝑝 ∈ 𝐺 ∣ dom (𝑝 ∖ I ) ∈ Fin}) |
10 | 9 | eqcomd 2801 | . . 3 ⊢ (𝐷 ∈ Fin → {𝑝 ∈ 𝐺 ∣ dom (𝑝 ∖ I ) ∈ Fin} = 𝐺) |
11 | 10 | fneq2d 6317 | . 2 ⊢ (𝐷 ∈ Fin → (𝑆 Fn {𝑝 ∈ 𝐺 ∣ dom (𝑝 ∖ I ) ∈ Fin} ↔ 𝑆 Fn 𝐺)) |
12 | 5, 11 | mpbii 234 | 1 ⊢ (𝐷 ∈ Fin → 𝑆 Fn 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 ∀wral 3105 {crab 3109 ∖ cdif 3856 I cid 5347 dom cdm 5443 Fn wfn 6220 ‘cfv 6225 Fincfn 8357 Basecbs 16312 SymGrpcsymg 18236 pmSgncpsgn 18348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-uz 12094 df-fz 12743 df-fzo 12884 df-hash 13541 df-word 13708 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-plusg 16407 df-tset 16413 df-symg 18237 df-psgn 18350 |
This theorem is referenced by: mdetpmtr1 30703 |
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