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 34991
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 2770 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2770 . . 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 4114 . . . 4 ∅ ⊆ (Base‘𝐾)
76a1i 11 . . 3 (𝐾 ∈ OP → ∅ ⊆ (Base‘𝐾))
81, 2, 3, 4, 5, 7lubval 17191 . 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 34986 . . 3 (𝐾 ∈ OP → 0 ∈ (Base‘𝐾))
11 ral0 4215 . . . . . . 7 𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧
1211a1bi 351 . . . . . 6 (𝑥(le‘𝐾)𝑧 ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))
1312ralbii 3128 . . . . 5 (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))
14 ral0 4215 . . . . . 6 𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥
1514biantrur 514 . . . . 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 466 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 0 ∈ (Base‘𝐾))
18 breq2 4788 . . . . . . . 8 (𝑧 = 0 → (𝑥(le‘𝐾)𝑧𝑥(le‘𝐾) 0 ))
1918rspcv 3454 . . . . . . 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 34989 . . . . . 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 34988 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
2423adantlr 686 . . . . . . . . 9 (((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
2524ex 397 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → 0 (le‘𝐾)𝑧))
26 breq1 4787 . . . . . . . . 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 3116 . . . . 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 6779 . 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 2804 1 (𝐾 ∈ OP → ( 1 ‘∅) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  wss 3721  c0 4061   class class class wbr 4784  cfv 6031  crio 6752  Basecbs 16063  lecple 16155  lubclub 17149  0.cp0 17244  OPcops 34974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-preset 17135  df-poset 17153  df-lub 17181  df-glb 17182  df-p0 17246  df-oposet 34978
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator