Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ddeval0 Structured version   Visualization version   GIF version

Theorem ddeval0 30603
Description: Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
Assertion
Ref Expression
ddeval0 ((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0)

Proof of Theorem ddeval0
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 reex 10215 . . . . 5 ℝ ∈ V
21ssex 4950 . . . 4 (𝐴 ⊆ ℝ → 𝐴 ∈ V)
3 elpwg 4306 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ))
43biimpar 503 . . . 4 ((𝐴 ∈ V ∧ 𝐴 ⊆ ℝ) → 𝐴 ∈ 𝒫 ℝ)
52, 4mpancom 706 . . 3 (𝐴 ⊆ ℝ → 𝐴 ∈ 𝒫 ℝ)
6 eleq2 2824 . . . . 5 (𝑎 = 𝐴 → (0 ∈ 𝑎 ↔ 0 ∈ 𝐴))
76ifbid 4248 . . . 4 (𝑎 = 𝐴 → if(0 ∈ 𝑎, 1, 0) = if(0 ∈ 𝐴, 1, 0))
8 df-dde 30601 . . . 4 δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
9 1ex 10223 . . . . 5 1 ∈ V
10 c0ex 10222 . . . . 5 0 ∈ V
119, 10ifex 4296 . . . 4 if(0 ∈ 𝐴, 1, 0) ∈ V
127, 8, 11fvmpt 6440 . . 3 (𝐴 ∈ 𝒫 ℝ → (δ‘𝐴) = if(0 ∈ 𝐴, 1, 0))
135, 12syl 17 . 2 (𝐴 ⊆ ℝ → (δ‘𝐴) = if(0 ∈ 𝐴, 1, 0))
14 iffalse 4235 . 2 (¬ 0 ∈ 𝐴 → if(0 ∈ 𝐴, 1, 0) = 0)
1513, 14sylan9eq 2810 1 ((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1628  wcel 2135  Vcvv 3336  wss 3711  ifcif 4226  𝒫 cpw 4298  cfv 6045  cr 10123  0cc0 10124  1c1 10125  δcdde 30600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-mulcl 10186  ax-i2m1 10192
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-iota 6008  df-fun 6047  df-fv 6053  df-dde 30601
This theorem is referenced by:  ddemeas  30604
  Copyright terms: Public domain W3C validator