Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hl0lt1N Structured version   Visualization version   GIF version

Theorem hl0lt1N 34195
 Description: Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
hl0lt1.s < = (lt‘𝐾)
hl0lt1.z 0 = (0.‘𝐾)
hl0lt1.u 1 = (1.‘𝐾)
Assertion
Ref Expression
hl0lt1N (𝐾 ∈ HL → 0 < 1 )

Proof of Theorem hl0lt1N
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 hl0lt1.s . . 3 < = (lt‘𝐾)
3 hl0lt1.z . . 3 0 = (0.‘𝐾)
4 hl0lt1.u . . 3 1 = (1.‘𝐾)
51, 2, 3, 4hlhgt2 34194 . 2 (𝐾 ∈ HL → ∃𝑥 ∈ (Base‘𝐾)( 0 < 𝑥𝑥 < 1 ))
6 hlpos 34171 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Poset)
76adantr 481 . . . 4 ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ Poset)
8 hlop 34168 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OP)
98adantr 481 . . . . 5 ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ OP)
101, 3op0cl 33990 . . . . 5 (𝐾 ∈ OP → 0 ∈ (Base‘𝐾))
119, 10syl 17 . . . 4 ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 0 ∈ (Base‘𝐾))
12 simpr 477 . . . 4 ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾))
131, 4op1cl 33991 . . . . 5 (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
149, 13syl 17 . . . 4 ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 1 ∈ (Base‘𝐾))
151, 2plttr 16910 . . . 4 ((𝐾 ∈ Poset ∧ ( 0 ∈ (Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 1 ∈ (Base‘𝐾))) → (( 0 < 𝑥𝑥 < 1 ) → 0 < 1 ))
167, 11, 12, 14, 15syl13anc 1325 . . 3 ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → (( 0 < 𝑥𝑥 < 1 ) → 0 < 1 ))
1716rexlimdva 3026 . 2 (𝐾 ∈ HL → (∃𝑥 ∈ (Base‘𝐾)( 0 < 𝑥𝑥 < 1 ) → 0 < 1 ))
185, 17mpd 15 1 (𝐾 ∈ HL → 0 < 1 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∃wrex 2909   class class class wbr 4623  ‘cfv 5857  Basecbs 15800  Posetcpo 16880  ltcplt 16881  0.cp0 16977  1.cp1 16978  OPcops 33978  HLchlt 34156 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-plt 16898  df-lub 16914  df-glb 16915  df-p0 16979  df-p1 16980  df-lat 16986  df-oposet 33982  df-ol 33984  df-oml 33985  df-atl 34104  df-cvlat 34128  df-hlat 34157 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator