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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)
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