MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  stdbdmetval Structured version   Visualization version   GIF version

Theorem stdbdmetval 22552
Description: Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypothesis
Ref Expression
stdbdmet.1 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))
Assertion
Ref Expression
stdbdmetval ((𝑅𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐶,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem stdbdmetval
StepHypRef Expression
1 ovex 6915 . . . 4 (𝐴𝐶𝐵) ∈ V
2 ifexg 4337 . . . 4 (((𝐴𝐶𝐵) ∈ V ∧ 𝑅𝑉) → if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅) ∈ V)
31, 2mpan 673 . . 3 (𝑅𝑉 → if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅) ∈ V)
4 oveq12 6892 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝐶𝑦) = (𝐴𝐶𝐵))
54breq1d 4865 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝐶𝑦) ≤ 𝑅 ↔ (𝐴𝐶𝐵) ≤ 𝑅))
65, 4ifbieq1d 4313 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
7 stdbdmet.1 . . . 4 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))
86, 7ovmpt2ga 7029 . . 3 ((𝐴𝑋𝐵𝑋 ∧ if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅) ∈ V) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
93, 8syl3an3 1198 . 2 ((𝐴𝑋𝐵𝑋𝑅𝑉) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
1093comr 1148 1 ((𝑅𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2157  Vcvv 3402  ifcif 4290   class class class wbr 4855  (class class class)co 6883  cmpt2 6885  cle 10369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-iota 6073  df-fun 6112  df-fv 6118  df-ov 6886  df-oprab 6887  df-mpt2 6888
This theorem is referenced by:  stdbdbl  22555
  Copyright terms: Public domain W3C validator