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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 30391 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 37063 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | 1 | 3ad2ant1 1135 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ Lat) |
3 | simp2 1139 | . . 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 36989 | . . . 4 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
7 | 6 | 3ad2ant3 1137 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐵) |
8 | eqid 2736 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
9 | cvr2.s | . . . 4 ⊢ < = (lt‘𝐾) | |
10 | cvr2.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
11 | 4, 8, 9, 10 | latnle 17933 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (¬ 𝑃(le‘𝐾)𝑋 ↔ 𝑋 < (𝑋 ∨ 𝑃))) |
12 | 2, 3, 7, 11 | syl3anc 1373 | . 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 37110 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑃(le‘𝐾)𝑋 ↔ 𝑋𝐶(𝑋 ∨ 𝑃))) |
15 | 12, 14 | bitr3d 284 | 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 209 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 lecple 16756 ltcplt 17769 joincjn 17772 Latclat 17891 ⋖ ccvr 36962 Atomscatm 36963 HLchlt 37050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-proset 17756 df-poset 17774 df-plt 17790 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-p0 17885 df-lat 17892 df-clat 17959 df-oposet 36876 df-ol 36878 df-oml 36879 df-covers 36966 df-ats 36967 df-atl 36998 df-cvlat 37022 df-hlat 37051 |
This theorem is referenced by: cvrval4N 37114 |
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