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

Theorem eqinfd 9525
Description: Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 3-Sep-2020.)
Hypotheses
Ref Expression
infexd.1 (𝜑𝑅 Or 𝐴)
eqinfd.2 (𝜑𝐶𝐴)
eqinfd.3 ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)
eqinfd.4 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
Assertion
Ref Expression
eqinfd (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧   𝑦,𝑅,𝑧   𝑦,𝐶,𝑧   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑧)

Proof of Theorem eqinfd
StepHypRef Expression
1 eqinfd.2 . 2 (𝜑𝐶𝐴)
2 eqinfd.3 . . 3 ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)
32ralrimiva 3146 . 2 (𝜑 → ∀𝑦𝐵 ¬ 𝑦𝑅𝐶)
4 eqinfd.4 . . . 4 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
54expr 456 . . 3 ((𝜑𝑦𝐴) → (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))
65ralrimiva 3146 . 2 (𝜑 → ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))
7 infexd.1 . . 3 (𝜑𝑅 Or 𝐴)
87eqinf 9524 . 2 (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))
91, 3, 6, 8mp3and 1466 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070   class class class wbr 5143   Or wor 5591  infcinf 9481
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-po 5592  df-so 5593  df-cnv 5693  df-iota 6514  df-riota 7388  df-sup 9482  df-inf 9483
This theorem is referenced by:  infmin  9534  xrinf0  13380  infmremnf  13385  infmrp1  13386
  Copyright terms: Public domain W3C validator