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Theorem glb0N 36438
Description: The greatest lower bound of the empty set is the unit element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
glb0.g 𝐺 = (glb‘𝐾)
glb0.u 1 = (1.‘𝐾)
Assertion
Ref Expression
glb0N (𝐾 ∈ OP → (𝐺‘∅) = 1 )

Proof of Theorem glb0N
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2824 . . 3 (le‘𝐾) = (le‘𝐾)
3 glb0.g . . 3 𝐺 = (glb‘𝐾)
4 biid 264 . . 3 ((∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)) ↔ (∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))
5 id 22 . . 3 (𝐾 ∈ OP → 𝐾 ∈ OP)
6 0ss 4333 . . . 4 ∅ ⊆ (Base‘𝐾)
76a1i 11 . . 3 (𝐾 ∈ OP → ∅ ⊆ (Base‘𝐾))
81, 2, 3, 4, 5, 7glbval 17607 . 2 (𝐾 ∈ OP → (𝐺‘∅) = (𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))))
9 glb0.u . . . 4 1 = (1.‘𝐾)
101, 9op1cl 36430 . . 3 (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
11 ral0 4439 . . . . . . 7 𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦
1211a1bi 366 . . . . . 6 (𝑧(le‘𝐾)𝑥 ↔ (∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))
1312ralbii 3160 . . . . 5 (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))
14 ral0 4439 . . . . . 6 𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦
1514biantrur 534 . . . . 5 (∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥) ↔ (∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))
1613, 15bitri 278 . . . 4 (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 ↔ (∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)))
1710adantr 484 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 1 ∈ (Base‘𝐾))
18 breq1 5055 . . . . . . . 8 (𝑧 = 1 → (𝑧(le‘𝐾)𝑥1 (le‘𝐾)𝑥))
1918rspcv 3604 . . . . . . 7 ( 1 ∈ (Base‘𝐾) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥1 (le‘𝐾)𝑥))
2017, 19syl 17 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥1 (le‘𝐾)𝑥))
211, 2, 9op1le 36437 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → ( 1 (le‘𝐾)𝑥𝑥 = 1 ))
2220, 21sylibd 242 . . . . 5 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥𝑥 = 1 ))
231, 2, 9ople1 36436 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧(le‘𝐾) 1 )
2423adantlr 714 . . . . . . . . 9 (((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧(le‘𝐾) 1 )
2524ex 416 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → 𝑧(le‘𝐾) 1 ))
26 breq2 5056 . . . . . . . . 9 (𝑥 = 1 → (𝑧(le‘𝐾)𝑥𝑧(le‘𝐾) 1 ))
2726biimprcd 253 . . . . . . . 8 (𝑧(le‘𝐾) 1 → (𝑥 = 1𝑧(le‘𝐾)𝑥))
2825, 27syl6 35 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → (𝑥 = 1𝑧(le‘𝐾)𝑥)))
2928com23 86 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 1 → (𝑧 ∈ (Base‘𝐾) → 𝑧(le‘𝐾)𝑥)))
3029ralrimdv 3183 . . . . 5 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 1 → ∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥))
3122, 30impbid 215 . . . 4 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥𝑥 = 1 ))
3216, 31bitr3id 288 . . 3 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → ((∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥)) ↔ 𝑥 = 1 ))
3310, 32riota5 7136 . 2 (𝐾 ∈ OP → (𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))) = 1 )
348, 33eqtrd 2859 1 (𝐾 ∈ OP → (𝐺‘∅) = 1 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  wss 3919  c0 4276   class class class wbr 5052  cfv 6343  crio 7106  Basecbs 16483  lecple 16572  glbcglb 17553  1.cp1 17648  OPcops 36417
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-proset 17538  df-poset 17556  df-lub 17584  df-glb 17585  df-p1 17650  df-oposet 36421
This theorem is referenced by:  pmapglb2N  37016  pmapglb2xN  37017
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