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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 38010 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑊) ∈ 𝐴) |
5 | lhpocnel.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
6 | 5, 1, 3 | lhpocnle 38009 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ¬ ( ⊥ ‘𝑊) ≤ 𝑊) |
7 | 4, 6 | jca 511 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( ⊥ ‘𝑊) ∈ 𝐴 ∧ ¬ ( ⊥ ‘𝑊) ≤ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 class class class wbr 5078 ‘cfv 6430 lecple 16950 occoc 16951 Atomscatm 37256 HLchlt 37343 LHypclh 37977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-proset 17994 df-poset 18012 df-plt 18029 df-lub 18045 df-glb 18046 df-meet 18048 df-p0 18124 df-p1 18125 df-lat 18131 df-oposet 37169 df-ol 37171 df-oml 37172 df-covers 37259 df-ats 37260 df-atl 37291 df-cvlat 37315 df-hlat 37344 df-lhyp 37981 |
This theorem is referenced by: lhpocnel2 38012 trlcl 38157 trlle 38177 cdlemk19w 38965 dia2dimlem8 39064 dicssdvh 39179 dicvaddcl 39183 dicvscacl 39184 dicn0 39185 dih1 39279 dihatlat 39327 |
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