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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpocnel | Structured version Visualization version GIF version |
Description: The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 25-May-2012.) |
Ref | Expression |
---|---|
lhpocnel.l | ⊢ ≤ = (le‘𝐾) |
lhpocnel.o | ⊢ ⊥ = (oc‘𝐾) |
lhpocnel.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhpocnel.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpocnel | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( ⊥ ‘𝑊) ∈ 𝐴 ∧ ¬ ( ⊥ ‘𝑊) ≤ 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpocnel.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
2 | lhpocnel.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | lhpocnel.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 1, 2, 3 | lhpocat 35303 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑊) ∈ 𝐴) |
5 | lhpocnel.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
6 | 5, 1, 3 | lhpocnle 35302 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ¬ ( ⊥ ‘𝑊) ≤ 𝑊) |
7 | 4, 6 | jca 554 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( ⊥ ‘𝑊) ∈ 𝐴 ∧ ¬ ( ⊥ ‘𝑊) ≤ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 lecple 15948 occoc 15949 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-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 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: lhpocnel2 35305 trlcl 35451 trlle 35471 cdlemk19w 36260 dia2dimlem8 36360 dicssdvh 36475 dicvaddcl 36479 dicvscacl 36480 dicn0 36481 dih1 36575 dihatlat 36623 |
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