Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 10631 | . . . . 5 ⊢ ℝ ∈ V | |
2 | 1 | ssex 5228 | . . . 4 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V) |
3 | elpwg 4545 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)) | |
4 | 3 | biimpar 480 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ ℝ) → 𝐴 ∈ 𝒫 ℝ) |
5 | 2, 4 | mpancom 686 | . . 3 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ 𝒫 ℝ) |
6 | eleq2 2904 | . . . . 5 ⊢ (𝑎 = 𝐴 → (0 ∈ 𝑎 ↔ 0 ∈ 𝐴)) | |
7 | 6 | ifbid 4492 | . . . 4 ⊢ (𝑎 = 𝐴 → if(0 ∈ 𝑎, 1, 0) = if(0 ∈ 𝐴, 1, 0)) |
8 | df-dde 31496 | . . . 4 ⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0)) | |
9 | 1ex 10640 | . . . . 5 ⊢ 1 ∈ V | |
10 | c0ex 10638 | . . . . 5 ⊢ 0 ∈ V | |
11 | 9, 10 | ifex 4518 | . . . 4 ⊢ if(0 ∈ 𝐴, 1, 0) ∈ V |
12 | 7, 8, 11 | fvmpt 6771 | . . 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 4479 | . 2 ⊢ (¬ 0 ∈ 𝐴 → if(0 ∈ 𝐴, 1, 0) = 0) | |
15 | 13, 14 | sylan9eq 2879 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 ifcif 4470 𝒫 cpw 4542 ‘cfv 6358 ℝcr 10539 0cc0 10540 1c1 10541 δcdde 31495 |
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 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-mulcl 10602 ax-i2m1 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-dde 31496 |
This theorem is referenced by: ddemeas 31499 |
Copyright terms: Public domain | W3C validator |