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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 3499 | . . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V) | |
3 | fveq2 6907 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
4 | 3 | cnveqd 5889 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) |
5 | 4 | imaeq1d 6079 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) |
6 | df-evpm 19525 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
7 | fvex 6920 | . . . . . 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 7016 | . . 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 2787 | 1 ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 ◡ccnv 5688 “ cima 5692 ‘cfv 6563 1c1 11154 pmSgncpsgn 19522 pmEvencevpm 19523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-evpm 19525 |
This theorem is referenced by: evpmsubg 33150 |
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