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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 2797 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
3 | lhp0lt.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 1, 2, 3 | lhpexlt 36015 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ (Atoms‘𝐾)𝑝 < 𝑊) |
5 | simp1l 1255 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝐾 ∈ HL) | |
6 | hlop 35375 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
7 | eqid 2797 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | lhp0lt.z | . . . . . . 7 ⊢ 0 = (0.‘𝐾) | |
9 | 7, 8 | op0cl 35197 | . . . . . 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 35302 | . . . . . 6 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾)) |
12 | 11 | 3ad2ant2 1165 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝑝 ∈ (Base‘𝐾)) |
13 | simp2 1168 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝑝 ∈ (Atoms‘𝐾)) | |
14 | eqid 2797 | . . . . . . 7 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
15 | 8, 14, 2 | atcvr0 35301 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾)) → 0 ( ⋖ ‘𝐾)𝑝) |
16 | 5, 13, 15 | syl2anc 580 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 0 ( ⋖ ‘𝐾)𝑝) |
17 | 7, 1, 14 | cvrlt 35283 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 0 ∈ (Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) ∧ 0 ( ⋖ ‘𝐾)𝑝) → 0 < 𝑝) |
18 | 5, 10, 12, 16, 17 | syl31anc 1493 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 0 < 𝑝) |
19 | simp3 1169 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝑝 < 𝑊) | |
20 | hlpos 35379 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) | |
21 | 5, 20 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝐾 ∈ Poset) |
22 | simp1r 1256 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 𝑊 ∈ 𝐻) | |
23 | 7, 3 | lhpbase 36011 | . . . . . 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 17282 | . . . . 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 1492 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → (( 0 < 𝑝 ∧ 𝑝 < 𝑊) → 0 < 𝑊)) |
27 | 18, 19, 26 | mp2and 691 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 < 𝑊) → 0 < 𝑊) |
28 | 27 | rexlimdv3a 3212 | . 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 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ∃wrex 3088 class class class wbr 4841 ‘cfv 6099 Basecbs 16181 Posetcpo 17252 ltcplt 17253 0.cp0 17349 OPcops 35185 ⋖ ccvr 35275 Atomscatm 35276 HLchlt 35363 LHypclh 35997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-proset 17240 df-poset 17258 df-plt 17270 df-lub 17286 df-glb 17287 df-join 17288 df-meet 17289 df-p0 17351 df-p1 17352 df-lat 17358 df-clat 17420 df-oposet 35189 df-ol 35191 df-oml 35192 df-covers 35279 df-ats 35280 df-atl 35311 df-cvlat 35335 df-hlat 35364 df-lhyp 36001 |
This theorem is referenced by: lhpn0 36017 |
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