Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  glb0N Structured version   Visualization version   GIF version

Theorem glb0N 38052
Description: The greatest lower bound of the empty set is the unity element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
glb0.g 𝐺 = (glbβ€˜πΎ)
glb0.u 1 = (1.β€˜πΎ)
Assertion
Ref Expression
glb0N (𝐾 ∈ OP β†’ (πΊβ€˜βˆ…) = 1 )

Proof of Theorem glb0N
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2733 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
3 glb0.g . . 3 𝐺 = (glbβ€˜πΎ)
4 biid 261 . . 3 ((βˆ€π‘¦ ∈ βˆ… π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ βˆ… 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)) ↔ (βˆ€π‘¦ ∈ βˆ… π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ βˆ… 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)))
5 id 22 . . 3 (𝐾 ∈ OP β†’ 𝐾 ∈ OP)
6 0ss 4396 . . . 4 βˆ… βŠ† (Baseβ€˜πΎ)
76a1i 11 . . 3 (𝐾 ∈ OP β†’ βˆ… βŠ† (Baseβ€˜πΎ))
81, 2, 3, 4, 5, 7glbval 18319 . 2 (𝐾 ∈ OP β†’ (πΊβ€˜βˆ…) = (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ βˆ… π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ βˆ… 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯))))
9 glb0.u . . . 4 1 = (1.β€˜πΎ)
101, 9op1cl 38044 . . 3 (𝐾 ∈ OP β†’ 1 ∈ (Baseβ€˜πΎ))
11 ral0 4512 . . . . . . 7 βˆ€π‘¦ ∈ βˆ… 𝑧(leβ€˜πΎ)𝑦
1211a1bi 363 . . . . . 6 (𝑧(leβ€˜πΎ)π‘₯ ↔ (βˆ€π‘¦ ∈ βˆ… 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯))
1312ralbii 3094 . . . . 5 (βˆ€π‘§ ∈ (Baseβ€˜πΎ)𝑧(leβ€˜πΎ)π‘₯ ↔ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ βˆ… 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯))
14 ral0 4512 . . . . . 6 βˆ€π‘¦ ∈ βˆ… π‘₯(leβ€˜πΎ)𝑦
1514biantrur 532 . . . . 5 (βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ βˆ… 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯) ↔ (βˆ€π‘¦ ∈ βˆ… π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ βˆ… 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)))
1613, 15bitri 275 . . . 4 (βˆ€π‘§ ∈ (Baseβ€˜πΎ)𝑧(leβ€˜πΎ)π‘₯ ↔ (βˆ€π‘¦ ∈ βˆ… π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ βˆ… 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)))
1710adantr 482 . . . . . . 7 ((𝐾 ∈ OP ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ 1 ∈ (Baseβ€˜πΎ))
18 breq1 5151 . . . . . . . 8 (𝑧 = 1 β†’ (𝑧(leβ€˜πΎ)π‘₯ ↔ 1 (leβ€˜πΎ)π‘₯))
1918rspcv 3609 . . . . . . 7 ( 1 ∈ (Baseβ€˜πΎ) β†’ (βˆ€π‘§ ∈ (Baseβ€˜πΎ)𝑧(leβ€˜πΎ)π‘₯ β†’ 1 (leβ€˜πΎ)π‘₯))
2017, 19syl 17 . . . . . 6 ((𝐾 ∈ OP ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (βˆ€π‘§ ∈ (Baseβ€˜πΎ)𝑧(leβ€˜πΎ)π‘₯ β†’ 1 (leβ€˜πΎ)π‘₯))
211, 2, 9op1le 38051 . . . . . 6 ((𝐾 ∈ OP ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ ( 1 (leβ€˜πΎ)π‘₯ ↔ π‘₯ = 1 ))
2220, 21sylibd 238 . . . . 5 ((𝐾 ∈ OP ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (βˆ€π‘§ ∈ (Baseβ€˜πΎ)𝑧(leβ€˜πΎ)π‘₯ β†’ π‘₯ = 1 ))
231, 2, 9ople1 38050 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑧 ∈ (Baseβ€˜πΎ)) β†’ 𝑧(leβ€˜πΎ) 1 )
2423adantlr 714 . . . . . . . . 9 (((𝐾 ∈ OP ∧ π‘₯ ∈ (Baseβ€˜πΎ)) ∧ 𝑧 ∈ (Baseβ€˜πΎ)) β†’ 𝑧(leβ€˜πΎ) 1 )
2524ex 414 . . . . . . . 8 ((𝐾 ∈ OP ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (𝑧 ∈ (Baseβ€˜πΎ) β†’ 𝑧(leβ€˜πΎ) 1 ))
26 breq2 5152 . . . . . . . . 9 (π‘₯ = 1 β†’ (𝑧(leβ€˜πΎ)π‘₯ ↔ 𝑧(leβ€˜πΎ) 1 ))
2726biimprcd 249 . . . . . . . 8 (𝑧(leβ€˜πΎ) 1 β†’ (π‘₯ = 1 β†’ 𝑧(leβ€˜πΎ)π‘₯))
2825, 27syl6 35 . . . . . . 7 ((𝐾 ∈ OP ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (𝑧 ∈ (Baseβ€˜πΎ) β†’ (π‘₯ = 1 β†’ 𝑧(leβ€˜πΎ)π‘₯)))
2928com23 86 . . . . . 6 ((𝐾 ∈ OP ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (π‘₯ = 1 β†’ (𝑧 ∈ (Baseβ€˜πΎ) β†’ 𝑧(leβ€˜πΎ)π‘₯)))
3029ralrimdv 3153 . . . . 5 ((𝐾 ∈ OP ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (π‘₯ = 1 β†’ βˆ€π‘§ ∈ (Baseβ€˜πΎ)𝑧(leβ€˜πΎ)π‘₯))
3122, 30impbid 211 . . . 4 ((𝐾 ∈ OP ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (βˆ€π‘§ ∈ (Baseβ€˜πΎ)𝑧(leβ€˜πΎ)π‘₯ ↔ π‘₯ = 1 ))
3216, 31bitr3id 285 . . 3 ((𝐾 ∈ OP ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ ((βˆ€π‘¦ ∈ βˆ… π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ βˆ… 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯)) ↔ π‘₯ = 1 ))
3310, 32riota5 7392 . 2 (𝐾 ∈ OP β†’ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ βˆ… π‘₯(leβ€˜πΎ)𝑦 ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ βˆ… 𝑧(leβ€˜πΎ)𝑦 β†’ 𝑧(leβ€˜πΎ)π‘₯))) = 1 )
348, 33eqtrd 2773 1 (𝐾 ∈ OP β†’ (πΊβ€˜βˆ…) = 1 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3948  βˆ…c0 4322   class class class wbr 5148  β€˜cfv 6541  β„©crio 7361  Basecbs 17141  lecple 17201  glbcglb 18260  1.cp1 18374  OPcops 38031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-proset 18245  df-poset 18263  df-lub 18296  df-glb 18297  df-p1 18376  df-oposet 38035
This theorem is referenced by:  pmapglb2N  38631  pmapglb2xN  38632
  Copyright terms: Public domain W3C validator