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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhp0lt | Structured version Visualization version GIF version |
Description: A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) |
Ref | Expression |
---|---|
lhp0lt.s | ⊢ < = (lt‘𝐾) |
lhp0lt.z | ⊢ 0 = (0.‘𝐾) |
lhp0lt.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhp0lt | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 < 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhp0lt.s | . . 3 ⊢ < = (lt‘𝐾) | |
2 | eqid 2610 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
3 | lhp0lt.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 1, 2, 3 | lhpexlt 34306 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ (Atoms‘𝐾)𝑝 < 𝑊) |
5 | simp1l 1078 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝐾 ∈ HL) | |
6 | hlop 33667 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
7 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | lhp0lt.z | . . . . . . 7 ⊢ 0 = (0.‘𝐾) | |
9 | 7, 8 | op0cl 33489 | . . . . . 6 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
10 | 5, 6, 9 | 3syl 18 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 0 ∈ (Base‘𝐾)) |
11 | 7, 2 | atbase 33594 | . . . . . 6 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾)) |
12 | 11 | 3ad2ant2 1076 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝑝 ∈ (Base‘𝐾)) |
13 | simp2 1055 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝑝 ∈ (Atoms‘𝐾)) | |
14 | eqid 2610 | . . . . . . 7 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
15 | 8, 14, 2 | atcvr0 33593 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾)) → 0 ( ⋖ ‘𝐾)𝑝) |
16 | 5, 13, 15 | syl2anc 691 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 0 ( ⋖ ‘𝐾)𝑝) |
17 | 7, 1, 14 | cvrlt 33575 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 0 ∈ (Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) ∧ 0 ( ⋖ ‘𝐾)𝑝) → 0 < 𝑝) |
18 | 5, 10, 12, 16, 17 | syl31anc 1321 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 0 < 𝑝) |
19 | simp3 1056 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝑝 < 𝑊) | |
20 | hlpos 33670 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) | |
21 | 5, 20 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝐾 ∈ Poset) |
22 | simp1r 1079 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝑊 ∈ 𝐻) | |
23 | 7, 3 | lhpbase 34302 | . . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
24 | 22, 23 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝑊 ∈ (Base‘𝐾)) |
25 | 7, 1 | plttr 16793 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ ( 0 ∈ (Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (( 0 < 𝑝 ∧ 𝑝 < 𝑊) → 0 < 𝑊)) |
26 | 21, 10, 12, 24, 25 | syl13anc 1320 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → (( 0 < 𝑝 ∧ 𝑝 < 𝑊) → 0 < 𝑊)) |
27 | 18, 19, 26 | mp2and 711 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 0 < 𝑊) |
28 | 27 | rexlimdv3a 3015 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (∃𝑝 ∈ (Atoms‘𝐾)𝑝 < 𝑊 → 0 < 𝑊)) |
29 | 4, 28 | mpd 15 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 < 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 Posetcpo 16763 ltcplt 16764 0.cp0 16860 OPcops 33477 ⋖ ccvr 33567 Atomscatm 33568 HLchlt 33655 LHypclh 34288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-lhyp 34292 |
This theorem is referenced by: lhpn0 34308 |
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