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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvr2N | Structured version Visualization version GIF version |
Description: Less-than and covers equivalence in a Hilbert lattice. (chcv2 29524 analog.) (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cvr2.b | ⊢ 𝐵 = (Base‘𝐾) |
cvr2.s | ⊢ < = (lt‘𝐾) |
cvr2.j | ⊢ ∨ = (join‘𝐾) |
cvr2.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
cvr2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
cvr2N | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 < (𝑋 ∨ 𝑃) ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 35153 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | 1 | 3ad2ant1 1128 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ Lat) |
3 | simp2 1132 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
4 | cvr2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | cvr2.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 4, 5 | atbase 35079 | . . . 4 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
7 | 6 | 3ad2ant3 1130 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐵) |
8 | eqid 2760 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
9 | cvr2.s | . . . 4 ⊢ < = (lt‘𝐾) | |
10 | cvr2.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
11 | 4, 8, 9, 10 | latnle 17286 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (¬ 𝑃(le‘𝐾)𝑋 ↔ 𝑋 < (𝑋 ∨ 𝑃))) |
12 | 2, 3, 7, 11 | syl3anc 1477 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃(le‘𝐾)𝑋 ↔ 𝑋 < (𝑋 ∨ 𝑃))) |
13 | cvr2.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
14 | 4, 8, 10, 13, 5 | cvr1 35199 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃(le‘𝐾)𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
15 | 12, 14 | bitr3d 270 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 < (𝑋 ∨ 𝑃) ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 lecple 16150 ltcplt 17142 joincjn 17145 Latclat 17246 ⋖ ccvr 35052 Atomscatm 35053 HLchlt 35140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p0 17240 df-lat 17247 df-clat 17309 df-oposet 34966 df-ol 34968 df-oml 34969 df-covers 35056 df-ats 35057 df-atl 35088 df-cvlat 35112 df-hlat 35141 |
This theorem is referenced by: cvrval4N 35203 |
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