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Theorem glb0N 37658
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 2737 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2737 . . 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 4357 . . . 4 βˆ… βŠ† (Baseβ€˜πΎ)
76a1i 11 . . 3 (𝐾 ∈ OP β†’ βˆ… βŠ† (Baseβ€˜πΎ))
81, 2, 3, 4, 5, 7glbval 18259 . 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 37650 . . 3 (𝐾 ∈ OP β†’ 1 ∈ (Baseβ€˜πΎ))
11 ral0 4471 . . . . . . 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 3097 . . . . 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 4471 . . . . . 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 5109 . . . . . . . 8 (𝑧 = 1 β†’ (𝑧(leβ€˜πΎ)π‘₯ ↔ 1 (leβ€˜πΎ)π‘₯))
1918rspcv 3578 . . . . . . 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 37657 . . . . . 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 37656 . . . . . . . . . 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 5110 . . . . . . . . 9 (π‘₯ = 1 β†’ (𝑧(leβ€˜πΎ)π‘₯ ↔ 𝑧(leβ€˜πΎ) 1 ))
2726biimprcd 250 . . . . . . . 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 3150 . . . . 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 7344 . 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 2777 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 3065   βŠ† wss 3911  βˆ…c0 4283   class class class wbr 5106  β€˜cfv 6497  β„©crio 7313  Basecbs 17084  lecple 17141  glbcglb 18200  1.cp1 18314  OPcops 37637
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-proset 18185  df-poset 18203  df-lub 18236  df-glb 18237  df-p1 18316  df-oposet 37641
This theorem is referenced by:  pmapglb2N  38237  pmapglb2xN  38238
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