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Theorem infglb 8636
Description: An infimum is the greatest lower bound. See also infcl 8634 and inflb 8635. (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
infglb (𝜑 → ((𝐶𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧𝐵 𝑧𝑅𝐶))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑧,𝐶   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem infglb
StepHypRef Expression
1 df-inf 8589 . . . . 5 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
21breq1i 4848 . . . 4 (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)
3 simpr 478 . . . . 5 ((𝜑𝐶𝐴) → 𝐶𝐴)
4 infcl.1 . . . . . . . 8 (𝜑𝑅 Or 𝐴)
5 cnvso 5891 . . . . . . . 8 (𝑅 Or 𝐴𝑅 Or 𝐴)
64, 5sylib 210 . . . . . . 7 (𝜑𝑅 Or 𝐴)
7 infcl.2 . . . . . . . 8 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
84, 7infcllem 8633 . . . . . . 7 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
96, 8supcl 8604 . . . . . 6 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
109adantr 473 . . . . 5 ((𝜑𝐶𝐴) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
11 brcnvg 5503 . . . . . 6 ((𝐶𝐴 ∧ sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
1211bicomd 215 . . . . 5 ((𝐶𝐴 ∧ sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) → (sup(𝐵, 𝐴, 𝑅)𝑅𝐶𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
133, 10, 12syl2anc 580 . . . 4 ((𝜑𝐶𝐴) → (sup(𝐵, 𝐴, 𝑅)𝑅𝐶𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
142, 13syl5bb 275 . . 3 ((𝜑𝐶𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
156, 8suplub 8606 . . . . 5 (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
1615expdimp 445 . . . 4 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧))
17 vex 3386 . . . . . 6 𝑧 ∈ V
18 brcnvg 5503 . . . . . 6 ((𝐶𝐴𝑧 ∈ V) → (𝐶𝑅𝑧𝑧𝑅𝐶))
193, 17, 18sylancl 581 . . . . 5 ((𝜑𝐶𝐴) → (𝐶𝑅𝑧𝑧𝑅𝐶))
2019rexbidv 3231 . . . 4 ((𝜑𝐶𝐴) → (∃𝑧𝐵 𝐶𝑅𝑧 ↔ ∃𝑧𝐵 𝑧𝑅𝐶))
2116, 20sylibd 231 . . 3 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑧𝑅𝐶))
2214, 21sylbid 232 . 2 ((𝜑𝐶𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 → ∃𝑧𝐵 𝑧𝑅𝐶))
2322expimpd 446 1 (𝜑 → ((𝐶𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧𝐵 𝑧𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wcel 2157  wral 3087  wrex 3088  Vcvv 3383   class class class wbr 4841   Or wor 5230  ccnv 5309  supcsup 8586  infcinf 8587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-po 5231  df-so 5232  df-cnv 5318  df-iota 6062  df-riota 6837  df-sup 8588  df-inf 8589
This theorem is referenced by:  infnlb  8638  omssubaddlem  30869  omssubadd  30870  gtinf  32818  infxrunb2  40316
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