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Mirrors > Home > MPE Home > Th. List > Mathboxes > ddeval0 | Structured version Visualization version GIF version |
Description: Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.) |
Ref | Expression |
---|---|
ddeval0 | ⊢ ((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reex 10820 | . . . . 5 ⊢ ℝ ∈ V | |
2 | 1 | ssex 5214 | . . . 4 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V) |
3 | elpwg 4516 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)) | |
4 | 3 | biimpar 481 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ ℝ) → 𝐴 ∈ 𝒫 ℝ) |
5 | 2, 4 | mpancom 688 | . . 3 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ 𝒫 ℝ) |
6 | eleq2 2826 | . . . . 5 ⊢ (𝑎 = 𝐴 → (0 ∈ 𝑎 ↔ 0 ∈ 𝐴)) | |
7 | 6 | ifbid 4462 | . . . 4 ⊢ (𝑎 = 𝐴 → if(0 ∈ 𝑎, 1, 0) = if(0 ∈ 𝐴, 1, 0)) |
8 | df-dde 31913 | . . . 4 ⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0)) | |
9 | 1ex 10829 | . . . . 5 ⊢ 1 ∈ V | |
10 | c0ex 10827 | . . . . 5 ⊢ 0 ∈ V | |
11 | 9, 10 | ifex 4489 | . . . 4 ⊢ if(0 ∈ 𝐴, 1, 0) ∈ V |
12 | 7, 8, 11 | fvmpt 6818 | . . 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 | iffalse 4448 | . 2 ⊢ (¬ 0 ∈ 𝐴 → if(0 ∈ 𝐴, 1, 0) = 0) | |
15 | 13, 14 | sylan9eq 2798 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ⊆ wss 3866 ifcif 4439 𝒫 cpw 4513 ‘cfv 6380 ℝcr 10728 0cc0 10729 1c1 10730 δcdde 31912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-mulcl 10791 ax-i2m1 10797 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-dde 31913 |
This theorem is referenced by: ddemeas 31916 |
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