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Theorem tosglb 31779
Description: Same theorem as toslub 31777, 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 31778 . . . 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 7332 . . 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 260 . . . 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 18257 . . 3 (𝜑 → ((glb‘𝐾)‘𝐴) = (𝑎𝐵 (∀𝑏𝐴 𝑎(le‘𝐾)𝑏 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑐(le‘𝐾)𝑏𝑐(le‘𝐾)𝑎))))
111, 5, 2tosso 18307 . . . . . . 7 (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
1211ibi 266 . . . . . 6 (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
1312simpld 495 . . . . 5 (𝐾 ∈ Toset → < Or 𝐵)
14 cnvso 6240 . . . . 5 ( < Or 𝐵 < Or 𝐵)
1513, 14sylib 217 . . . 4 (𝐾 ∈ Toset → < Or 𝐵)
16 id 22 . . . . 5 ( < Or 𝐵 < Or 𝐵)
1716supval2 9390 . . . 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 9378 . . . 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 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  wss 3910   class class class wbr 5105   I cid 5530   Or wor 5544  ccnv 5632  cres 5635  cfv 6496  crio 7311  supcsup 9375  infcinf 9376  Basecbs 17082  lecple 17139  ltcplt 18196  glbcglb 18198  Tosetctos 18304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7312  df-sup 9377  df-inf 9378  df-proset 18183  df-poset 18201  df-plt 18218  df-glb 18235  df-toset 18305
This theorem is referenced by:  xrsp0  31816
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