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Theorem glb0N 34299
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 2620 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2620 . . 3 (le‘𝐾) = (le‘𝐾)
3 glb0.g . . 3 𝐺 = (glb‘𝐾)
4 biid 251 . . 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 3963 . . . 4 ∅ ⊆ (Base‘𝐾)
76a1i 11 . . 3 (𝐾 ∈ OP → ∅ ⊆ (Base‘𝐾))
81, 2, 3, 4, 5, 7glbval 16978 . 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 34291 . . 3 (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
11 ral0 4067 . . . . . . 7 𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦
1211a1bi 352 . . . . . 6 (𝑧(le‘𝐾)𝑥 ↔ (∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))
1312ralbii 2977 . . . . 5 (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦𝑧(le‘𝐾)𝑥))
14 ral0 4067 . . . . . 6 𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦
1514biantrur 527 . . . . 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 264 . . . 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 481 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 1 ∈ (Base‘𝐾))
18 breq1 4647 . . . . . . . 8 (𝑧 = 1 → (𝑧(le‘𝐾)𝑥1 (le‘𝐾)𝑥))
1918rspcv 3300 . . . . . . 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 34298 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → ( 1 (le‘𝐾)𝑥𝑥 = 1 ))
2220, 21sylibd 229 . . . . 5 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥𝑥 = 1 ))
231, 2, 9ople1 34297 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧(le‘𝐾) 1 )
2423adantlr 750 . . . . . . . . 9 (((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧(le‘𝐾) 1 )
2524ex 450 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → 𝑧(le‘𝐾) 1 ))
26 breq2 4648 . . . . . . . . 9 (𝑥 = 1 → (𝑧(le‘𝐾)𝑥𝑧(le‘𝐾) 1 ))
2726biimprcd 240 . . . . . . . 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 2965 . . . . 5 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 1 → ∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥))
3122, 30impbid 202 . . . 4 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥𝑥 = 1 ))
3216, 31syl5bbr 274 . . 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 6622 . 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 2654 1 (𝐾 ∈ OP → (𝐺‘∅) = 1 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  wss 3567  c0 3907   class class class wbr 4644  cfv 5876  crio 6595  Basecbs 15838  lecple 15929  glbcglb 16924  1.cp1 17019  OPcops 34278
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-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-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-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-ov 6638  df-preset 16909  df-poset 16927  df-lub 16955  df-glb 16956  df-p1 17021  df-oposet 34282
This theorem is referenced by:  pmapglb2N  34876  pmapglb2xN  34877
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