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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlatcvr1 | Structured version Visualization version GIF version |
Description: An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
cvlatcvr1.j | ⊢ ∨ = (join‘𝐾) |
cvlatcvr1.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
cvlatcvr1.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
cvlatcvr1 | ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ 𝑃𝐶(𝑃 ∨ 𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp13 1202 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ CvLat) | |
2 | cvlatl 36621 | . . . 4 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ AtLat) |
4 | eqid 2798 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
5 | eqid 2798 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
6 | cvlatcvr1.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | 4, 5, 6 | atnem0 36614 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ (𝑃(meet‘𝐾)𝑄) = (0.‘𝐾))) |
8 | 3, 7 | syld3an1 1407 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ (𝑃(meet‘𝐾)𝑄) = (0.‘𝐾))) |
9 | eqid 2798 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
10 | 9, 6 | atbase 36585 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
11 | cvlatcvr1.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
12 | cvlatcvr1.c | . . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
13 | 9, 11, 4, 5, 12, 6 | cvlcvrp 36636 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ 𝐴) → ((𝑃(meet‘𝐾)𝑄) = (0.‘𝐾) ↔ 𝑃𝐶(𝑃 ∨ 𝑄))) |
14 | 10, 13 | syl3an2 1161 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃(meet‘𝐾)𝑄) = (0.‘𝐾) ↔ 𝑃𝐶(𝑃 ∨ 𝑄))) |
15 | 8, 14 | bitrd 282 | 1 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ 𝑃𝐶(𝑃 ∨ 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 joincjn 17546 meetcmee 17547 0.cp0 17639 CLatccla 17709 OMLcoml 36471 ⋖ ccvr 36558 Atomscatm 36559 AtLatcal 36560 CvLatclc 36561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-lat 17648 df-clat 17710 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 |
This theorem is referenced by: cvlatcvr2 36638 atcvr1 36713 |
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