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Theorem tosglb 32966
Description: Same theorem as toslub 32964, 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 2736 . . . . 5 (le‘𝐾) = (le‘𝐾)
61, 2, 3, 4, 5tosglblem 32965 . . . 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 7408 . . 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 2736 . . . 4 (glb‘𝐾) = (glb‘𝐾)
9 biid 261 . . . 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 18415 . . 3 (𝜑 → ((glb‘𝐾)‘𝐴) = (𝑎𝐵 (∀𝑏𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑐(le‘𝐾)𝑏𝑐(le‘𝐾)𝑎))))
111, 5, 2tosso 18465 . . . . . . 7 (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
1211ibi 267 . . . . . 6 (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
1312simpld 494 . . . . 5 (𝐾 ∈ Toset → < Or 𝐵)
14 cnvso 6307 . . . . 5 ( < Or 𝐵 < Or 𝐵)
1513, 14sylib 218 . . . 4 (𝐾 ∈ Toset → < Or 𝐵)
16 id 22 . . . . 5 ( < Or 𝐵 < Or 𝐵)
1716supval2 9496 . . . 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 2786 . 2 (𝜑 → ((glb‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))
20 df-inf 9484 . . . 4 inf(𝐴, 𝐵, < ) = sup(𝐴, 𝐵, < )
2120eqcomi 2745 . . 3 sup(𝐴, 𝐵, < ) = inf(𝐴, 𝐵, < )
2221a1i 11 . 2 (𝜑 → sup(𝐴, 𝐵, < ) = inf(𝐴, 𝐵, < ))
2319, 22eqtrd 2776 1 (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  wrex 3069  wss 3950   class class class wbr 5142   I cid 5576   Or wor 5590  ccnv 5683  cres 5686  cfv 6560  crio 7388  supcsup 9481  infcinf 9482  Basecbs 17248  lecple 17305  ltcplt 18355  glbcglb 18357  Tosetctos 18462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-po 5591  df-so 5592  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-sup 9483  df-inf 9484  df-proset 18341  df-poset 18360  df-plt 18376  df-glb 18393  df-toset 18463
This theorem is referenced by:  xrsp0  33015
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