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| 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 3500 | . . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V) | |
| 3 | fveq2 6905 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
| 4 | 3 | cnveqd 5885 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) | 
| 5 | 4 | imaeq1d 6076 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) | 
| 6 | df-evpm 19511 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
| 7 | fvex 6918 | . . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V | |
| 8 | 7 | cnvex 7948 | . . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V | 
| 9 | 8 | imaex 7937 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V | 
| 10 | 5, 6, 9 | fvmpt 7015 | . . 3 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1})) | 
| 11 | 2, 10 | syl 17 | . 2 ⊢ (𝐷 ∈ 𝑉 → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1})) | 
| 12 | 1, 11 | eqtrid 2788 | 1 ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 {csn 4625 ◡ccnv 5683 “ cima 5687 ‘cfv 6560 1c1 11157 pmSgncpsgn 19508 pmEvencevpm 19509 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-evpm 19511 | 
| This theorem is referenced by: evpmsubg 33168 | 
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