Mathbox for Thierry Arnoux |
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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 3509 | . . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V) | |
3 | fveq2 6663 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
4 | 3 | cnveqd 5739 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) |
5 | 4 | imaeq1d 5921 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) |
6 | df-evpm 18615 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
7 | fvex 6676 | . . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V | |
8 | 7 | cnvex 7623 | . . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V |
9 | 8 | imaex 7614 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V |
10 | 5, 6, 9 | fvmpt 6761 | . . 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 2867 | 1 ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3491 {csn 4560 ◡ccnv 5547 “ cima 5551 ‘cfv 6348 1c1 10531 pmSgncpsgn 18612 pmEvencevpm 18613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-evpm 18615 |
This theorem is referenced by: evpmsubg 30810 |
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