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Theorem tosglb 29001
 Description: Same theorem as toslub 28999, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by AV, 28-Sep-2020.)
Hypotheses
Ref Expression
tosglb.b 𝐵 = (Base‘𝐾)
tosglb.l < = (lt‘𝐾)
tosglb.1 (𝜑𝐾 ∈ Toset)
tosglb.2 (𝜑𝐴𝐵)
Assertion
Ref Expression
tosglb (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < ))

Proof of Theorem tosglb
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tosglb.b . . . . 5 𝐵 = (Base‘𝐾)
2 tosglb.l . . . . 5 < = (lt‘𝐾)
3 tosglb.1 . . . . 5 (𝜑𝐾 ∈ Toset)
4 tosglb.2 . . . . 5 (𝜑𝐴𝐵)
5 eqid 2610 . . . . 5 (le‘𝐾) = (le‘𝐾)
61, 2, 3, 4, 5tosglblem 29000 . . . 4 ((𝜑𝑎𝐵) → ((∀𝑏𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑐(le‘𝐾)𝑏𝑐(le‘𝐾)𝑎)) ↔ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))
76riotabidva 6527 . . 3 (𝜑 → (𝑎𝐵 (∀𝑏𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑐(le‘𝐾)𝑏𝑐(le‘𝐾)𝑎))) = (𝑎𝐵 (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))
8 eqid 2610 . . . 4 (glb‘𝐾) = (glb‘𝐾)
9 biid 250 . . . 4 ((∀𝑏𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑐(le‘𝐾)𝑏𝑐(le‘𝐾)𝑎)) ↔ (∀𝑏𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑐(le‘𝐾)𝑏𝑐(le‘𝐾)𝑎)))
101, 5, 8, 9, 3, 4glbval 16820 . . 3 (𝜑 → ((glb‘𝐾)‘𝐴) = (𝑎𝐵 (∀𝑏𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑐(le‘𝐾)𝑏𝑐(le‘𝐾)𝑎))))
111, 5, 2tosso 16859 . . . . . . 7 (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
1211ibi 255 . . . . . 6 (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
1312simpld 474 . . . . 5 (𝐾 ∈ Toset → < Or 𝐵)
14 cnvso 5591 . . . . 5 ( < Or 𝐵 < Or 𝐵)
1513, 14sylib 207 . . . 4 (𝐾 ∈ Toset → < Or 𝐵)
16 id 22 . . . . 5 ( < Or 𝐵 < Or 𝐵)
1716supval2 8244 . . . 4 ( < Or 𝐵 → sup(𝐴, 𝐵, < ) = (𝑎𝐵 (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))
183, 15, 173syl 18 . . 3 (𝜑 → sup(𝐴, 𝐵, < ) = (𝑎𝐵 (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))
197, 10, 183eqtr4d 2654 . 2 (𝜑 → ((glb‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))
20 df-inf 8232 . . . 4 inf(𝐴, 𝐵, < ) = sup(𝐴, 𝐵, < )
2120eqcomi 2619 . . 3 sup(𝐴, 𝐵, < ) = inf(𝐴, 𝐵, < )
2221a1i 11 . 2 (𝜑 → sup(𝐴, 𝐵, < ) = inf(𝐴, 𝐵, < ))
2319, 22eqtrd 2644 1 (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   class class class wbr 4583   I cid 4948   Or wor 4958  ◡ccnv 5037   ↾ cres 5040  ‘cfv 5804  ℩crio 6510  supcsup 8229  infcinf 8230  Basecbs 15695  lecple 15775  ltcplt 16764  glbcglb 16766  Tosetctos 16856 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-sup 8231  df-inf 8232  df-preset 16751  df-poset 16769  df-plt 16781  df-glb 16798  df-toset 16857 This theorem is referenced by:  xrsp0  29012
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