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Theorem tosglb 29644
Description: Same theorem as toslub 29642, 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 2620 . . . . 5 (le‘𝐾) = (le‘𝐾)
61, 2, 3, 4, 5tosglblem 29643 . . . 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 6612 . . 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 2620 . . . 4 (glb‘𝐾) = (glb‘𝐾)
9 biid 251 . . . 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 16978 . . 3 (𝜑 → ((glb‘𝐾)‘𝐴) = (𝑎𝐵 (∀𝑏𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑐(le‘𝐾)𝑏𝑐(le‘𝐾)𝑎))))
111, 5, 2tosso 17017 . . . . . . 7 (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
1211ibi 256 . . . . . 6 (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
1312simpld 475 . . . . 5 (𝐾 ∈ Toset → < Or 𝐵)
14 cnvso 5662 . . . . 5 ( < Or 𝐵 < Or 𝐵)
1513, 14sylib 208 . . . 4 (𝐾 ∈ Toset → < Or 𝐵)
16 id 22 . . . . 5 ( < Or 𝐵 < Or 𝐵)
1716supval2 8346 . . . 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 2664 . 2 (𝜑 → ((glb‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))
20 df-inf 8334 . . . 4 inf(𝐴, 𝐵, < ) = sup(𝐴, 𝐵, < )
2120eqcomi 2629 . . 3 sup(𝐴, 𝐵, < ) = inf(𝐴, 𝐵, < )
2221a1i 11 . 2 (𝜑 → sup(𝐴, 𝐵, < ) = inf(𝐴, 𝐵, < ))
2319, 22eqtrd 2654 1 (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  wrex 2910  wss 3567   class class class wbr 4644   I cid 5013   Or wor 5024  ccnv 5103  cres 5106  cfv 5876  crio 6595  supcsup 8331  infcinf 8332  Basecbs 15838  lecple 15929  ltcplt 16922  glbcglb 16924  Tosetctos 17014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-po 5025  df-so 5026  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-sup 8333  df-inf 8334  df-preset 16909  df-poset 16927  df-plt 16939  df-glb 16956  df-toset 17015
This theorem is referenced by:  xrsp0  29655
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