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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 790 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL) | |
2 | simprl 794 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴) | |
3 | eqid 2622 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | lhplt.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 3, 4 | lhpbase 35284 | . . 3 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
6 | 5 | ad2antlr 763 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾)) |
7 | eqid 2622 | . . . 4 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
8 | eqid 2622 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | 7, 8, 4 | lhp1cvr 35285 | . . 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 796 | . 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 34760 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑊 ∈ (Base‘𝐾)) ∧ (𝑊( ⋖ ‘𝐾)(1.‘𝐾) ∧ 𝑃 ≤ 𝑊)) → 𝑃 < 𝑊) |
16 | 1, 2, 6, 10, 11, 15 | syl32anc 1334 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) → 𝑃 < 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 Basecbs 15857 lecple 15948 ltcplt 16941 1.cp1 17038 ⋖ ccvr 34549 Atomscatm 34550 HLchlt 34637 LHypclh 35270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-lhyp 35274 |
This theorem is referenced by: lhpexle1 35294 |
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