![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > evpmval | Structured version Visualization version GIF version |
Description: Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.) |
Ref | Expression |
---|---|
evpmval.1 | ⊢ 𝐴 = (pmEven‘𝐷) |
Ref | Expression |
---|---|
evpmval | ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evpmval.1 | . 2 ⊢ 𝐴 = (pmEven‘𝐷) | |
2 | elex 3459 | . . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V) | |
3 | fveq2 6645 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
4 | 3 | cnveqd 5710 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) |
5 | 4 | imaeq1d 5895 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) |
6 | df-evpm 18612 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
7 | fvex 6658 | . . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V | |
8 | 7 | cnvex 7612 | . . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V |
9 | 8 | imaex 7603 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V |
10 | 5, 6, 9 | fvmpt 6745 | . . 3 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1})) |
11 | 2, 10 | syl 17 | . 2 ⊢ (𝐷 ∈ 𝑉 → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1})) |
12 | 1, 11 | syl5eq 2845 | 1 ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 ◡ccnv 5518 “ cima 5522 ‘cfv 6324 1c1 10527 pmSgncpsgn 18609 pmEvencevpm 18610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 df-evpm 18612 |
This theorem is referenced by: evpmsubg 30839 |
Copyright terms: Public domain | W3C validator |