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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 11223 | . . . . 5 ⊢ ℝ ∈ V | |
2 | 1 | ssex 5315 | . . . 4 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V) |
3 | elpwg 4601 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)) | |
4 | 3 | biimpar 477 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ ℝ) → 𝐴 ∈ 𝒫 ℝ) |
5 | 2, 4 | mpancom 687 | . . 3 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ 𝒫 ℝ) |
6 | eleq2 2817 | . . . . 5 ⊢ (𝑎 = 𝐴 → (0 ∈ 𝑎 ↔ 0 ∈ 𝐴)) | |
7 | 6 | ifbid 4547 | . . . 4 ⊢ (𝑎 = 𝐴 → if(0 ∈ 𝑎, 1, 0) = if(0 ∈ 𝐴, 1, 0)) |
8 | df-dde 33842 | . . . 4 ⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0)) | |
9 | 1ex 11234 | . . . . 5 ⊢ 1 ∈ V | |
10 | c0ex 11232 | . . . . 5 ⊢ 0 ∈ V | |
11 | 9, 10 | ifex 4574 | . . . 4 ⊢ if(0 ∈ 𝐴, 1, 0) ∈ V |
12 | 7, 8, 11 | fvmpt 6999 | . . 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 4530 | . 2 ⊢ (0 ∈ 𝐴 → if(0 ∈ 𝐴, 1, 0) = 1) | |
15 | 13, 14 | sylan9eq 2787 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ 𝐴) → (δ‘𝐴) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ⊆ wss 3944 ifcif 4524 𝒫 cpw 4598 ‘cfv 6542 ℝcr 11131 0cc0 11132 1c1 11133 δcdde 33841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-mulcl 11194 ax-i2m1 11200 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-dde 33842 |
This theorem is referenced by: ddemeas 33845 |
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