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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhplt | Structured version Visualization version GIF version |
Description: An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) |
Ref | Expression |
---|---|
lhplt.l | ⊢ ≤ = (le‘𝐾) |
lhplt.s | ⊢ < = (lt‘𝐾) |
lhplt.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhplt.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhplt | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) → 𝑃 < 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 765 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL) | |
2 | simprl 769 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴) | |
3 | eqid 2737 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | lhplt.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 3, 4 | lhpbase 38399 | . . 3 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
6 | 5 | ad2antlr 725 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾)) |
7 | eqid 2737 | . . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
8 | eqid 2737 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | 7, 8, 4 | lhp1cvr 38400 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊( ⋖ ‘𝐾)(1.‘𝐾)) |
10 | 9 | adantr 481 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) → 𝑊( ⋖ ‘𝐾)(1.‘𝐾)) |
11 | simprr 771 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) → 𝑃 ≤ 𝑊) | |
12 | lhplt.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
13 | lhplt.s | . . 3 ⊢ < = (lt‘𝐾) | |
14 | lhplt.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
15 | 3, 12, 13, 7, 8, 14 | 1cvratlt 37875 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑊 ∈ (Base‘𝐾)) ∧ (𝑊( ⋖ ‘𝐾)(1.‘𝐾) ∧ 𝑃 ≤ 𝑊)) → 𝑃 < 𝑊) |
16 | 1, 2, 6, 10, 11, 15 | syl32anc 1378 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) → 𝑃 < 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ‘cfv 6493 Basecbs 17043 lecple 17100 ltcplt 18157 1.cp1 18273 ⋖ ccvr 37662 Atomscatm 37663 HLchlt 37750 LHypclh 38385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-proset 18144 df-poset 18162 df-plt 18179 df-lub 18195 df-glb 18196 df-join 18197 df-meet 18198 df-p0 18274 df-p1 18275 df-lat 18281 df-clat 18348 df-oposet 37576 df-ol 37578 df-oml 37579 df-covers 37666 df-ats 37667 df-atl 37698 df-cvlat 37722 df-hlat 37751 df-lhyp 38389 |
This theorem is referenced by: lhpexle1 38409 |
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