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

Theorem inflb 8550
Description: An infimum is a lower bound. See also infcl 8549 and infglb 8551. (Contributed by AV, 3-Sep-2020.)
Hypotheses
Ref Expression
infcl.1 (𝜑𝑅 Or 𝐴)
infcl.2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
Assertion
Ref Expression
inflb (𝜑 → (𝐶𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem inflb
StepHypRef Expression
1 infcl.1 . . . . . 6 (𝜑𝑅 Or 𝐴)
2 cnvso 5818 . . . . . 6 (𝑅 Or 𝐴𝑅 Or 𝐴)
31, 2sylib 208 . . . . 5 (𝜑𝑅 Or 𝐴)
4 infcl.2 . . . . . 6 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
51, 4infcllem 8548 . . . . 5 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
63, 5supub 8520 . . . 4 (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
76imp 393 . . 3 ((𝜑𝐶𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)
8 df-inf 8504 . . . . . 6 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
98a1i 11 . . . . 5 ((𝜑𝐶𝐵) → inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅))
109breq2d 4796 . . . 4 ((𝜑𝐶𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
113, 5supcl 8519 . . . . 5 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
12 brcnvg 5441 . . . . . 6 ((sup(𝐵, 𝐴, 𝑅) ∈ 𝐴𝐶𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝐶𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1312bicomd 213 . . . . 5 ((sup(𝐵, 𝐴, 𝑅) ∈ 𝐴𝐶𝐵) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
1411, 13sylan 561 . . . 4 ((𝜑𝐶𝐵) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
1510, 14bitrd 268 . . 3 ((𝜑𝐶𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
167, 15mtbird 314 . 2 ((𝜑𝐶𝐵) → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))
1716ex 397 1 (𝜑 → (𝐶𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wral 3060  wrex 3061   class class class wbr 4784   Or wor 5169  ccnv 5248  supcsup 8501  infcinf 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-po 5170  df-so 5171  df-cnv 5257  df-iota 5994  df-riota 6753  df-sup 8503  df-inf 8504
This theorem is referenced by:  infrelb  11209  infxrlb  12368  infssd  29822  infxrge0lb  29863  omssubadd  30696  ballotlemimin  30901  ballotlemfrcn0  30925  wsuclb  32104
  Copyright terms: Public domain W3C validator