Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lub0N Structured version   Visualization version   GIF version

Theorem lub0N 33995
Description: The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lub0.u 1 = (lub‘𝐾)
lub0.z 0 = (0.‘𝐾)
Assertion
Ref Expression
lub0N (𝐾 ∈ OP → ( 1 ‘∅) = 0 )

Proof of Theorem lub0N
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2621 . . 3 (le‘𝐾) = (le‘𝐾)
3 lub0.u . . 3 1 = (lub‘𝐾)
4 biid 251 . . 3 ((∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
5 id 22 . . 3 (𝐾 ∈ OP → 𝐾 ∈ OP)
6 0ss 3950 . . . 4 ∅ ⊆ (Base‘𝐾)
76a1i 11 . . 3 (𝐾 ∈ OP → ∅ ⊆ (Base‘𝐾))
81, 2, 3, 4, 5, 7lubval 16924 . 2 (𝐾 ∈ OP → ( 1 ‘∅) = (𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))))
9 lub0.z . . . 4 0 = (0.‘𝐾)
101, 9op0cl 33990 . . 3 (𝐾 ∈ OP → 0 ∈ (Base‘𝐾))
11 ral0 4054 . . . . . . 7 𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧
1211a1bi 352 . . . . . 6 (𝑥(le‘𝐾)𝑧 ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))
1312ralbii 2976 . . . . 5 (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))
14 ral0 4054 . . . . . 6 𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥
1514biantrur 527 . . . . 5 (∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧) ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
1613, 15bitri 264 . . . 4 (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
1710adantr 481 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 0 ∈ (Base‘𝐾))
18 breq2 4627 . . . . . . . 8 (𝑧 = 0 → (𝑥(le‘𝐾)𝑧𝑥(le‘𝐾) 0 ))
1918rspcv 3295 . . . . . . 7 ( 0 ∈ (Base‘𝐾) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧𝑥(le‘𝐾) 0 ))
2017, 19syl 17 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧𝑥(le‘𝐾) 0 ))
211, 2, 9ople0 33993 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥(le‘𝐾) 0𝑥 = 0 ))
2220, 21sylibd 229 . . . . 5 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧𝑥 = 0 ))
231, 2, 9op0le 33992 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
2423adantlr 750 . . . . . . . . 9 (((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
2524ex 450 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → 0 (le‘𝐾)𝑧))
26 breq1 4626 . . . . . . . . 9 (𝑥 = 0 → (𝑥(le‘𝐾)𝑧0 (le‘𝐾)𝑧))
2726biimprcd 240 . . . . . . . 8 ( 0 (le‘𝐾)𝑧 → (𝑥 = 0𝑥(le‘𝐾)𝑧))
2825, 27syl6 35 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → (𝑥 = 0𝑥(le‘𝐾)𝑧)))
2928com23 86 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 0 → (𝑧 ∈ (Base‘𝐾) → 𝑥(le‘𝐾)𝑧)))
3029ralrimdv 2964 . . . . 5 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 0 → ∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧))
3122, 30impbid 202 . . . 4 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧𝑥 = 0 ))
3216, 31syl5bbr 274 . . 3 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → ((∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)) ↔ 𝑥 = 0 ))
3310, 32riota5 6602 . 2 (𝐾 ∈ OP → (𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))) = 0 )
348, 33eqtrd 2655 1 (𝐾 ∈ OP → ( 1 ‘∅) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2908  wss 3560  c0 3897   class class class wbr 4623  cfv 5857  crio 6575  Basecbs 15800  lecple 15888  lubclub 16882  0.cp0 16977  OPcops 33978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-preset 16868  df-poset 16886  df-lub 16914  df-glb 16915  df-p0 16979  df-oposet 33982
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator