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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ddeval1 | Structured version Visualization version GIF version |
Description: Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.) |
Ref | Expression |
---|---|
ddeval1 | ⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reex 11200 | . . . . 5 ⊢ ℝ ∈ V | |
2 | 1 | ssex 5314 | . . . 4 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V) |
3 | elpwg 4600 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)) | |
4 | 3 | biimpar 477 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ ℝ) → 𝐴 ∈ 𝒫 ℝ) |
5 | 2, 4 | mpancom 685 | . . 3 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ 𝒫 ℝ) |
6 | eleq2 2816 | . . . . 5 ⊢ (𝑎 = 𝐴 → (0 ∈ 𝑎 ↔ 0 ∈ 𝐴)) | |
7 | 6 | ifbid 4546 | . . . 4 ⊢ (𝑎 = 𝐴 → if(0 ∈ 𝑎, 1, 0) = if(0 ∈ 𝐴, 1, 0)) |
8 | df-dde 33761 | . . . 4 ⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0)) | |
9 | 1ex 11211 | . . . . 5 ⊢ 1 ∈ V | |
10 | c0ex 11209 | . . . . 5 ⊢ 0 ∈ V | |
11 | 9, 10 | ifex 4573 | . . . 4 ⊢ if(0 ∈ 𝐴, 1, 0) ∈ V |
12 | 7, 8, 11 | fvmpt 6991 | . . 3 ⊢ (𝐴 ∈ 𝒫 ℝ → (δ‘𝐴) = if(0 ∈ 𝐴, 1, 0)) |
13 | 5, 12 | syl 17 | . 2 ⊢ (𝐴 ⊆ ℝ → (δ‘𝐴) = if(0 ∈ 𝐴, 1, 0)) |
14 | iftrue 4529 | . 2 ⊢ (0 ∈ 𝐴 → if(0 ∈ 𝐴, 1, 0) = 1) | |
15 | 13, 14 | sylan9eq 2786 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 ifcif 4523 𝒫 cpw 4597 ‘cfv 6536 ℝcr 11108 0cc0 11109 1c1 11110 δcdde 33760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-mulcl 11171 ax-i2m1 11177 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-dde 33761 |
This theorem is referenced by: ddemeas 33764 |
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