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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meetat2 | Structured version Visualization version GIF version |
Description: The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.) |
Ref | Expression |
---|---|
m.b | ⊢ 𝐵 = (Base‘𝐾) |
m.m | ⊢ ∧ = (meet‘𝐾) |
m.z | ⊢ 0 = (0.‘𝐾) |
m.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
meetat2 | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) ∈ 𝐴 ∨ (𝑋 ∧ 𝑃) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | m.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | m.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
3 | m.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
4 | m.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 1, 2, 3, 4 | meetat 38104 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 𝑃 ∨ (𝑋 ∧ 𝑃) = 0 )) |
6 | eleq1a 2829 | . . . 4 ⊢ (𝑃 ∈ 𝐴 → ((𝑋 ∧ 𝑃) = 𝑃 → (𝑋 ∧ 𝑃) ∈ 𝐴)) | |
7 | 6 | 3ad2ant3 1136 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 𝑃 → (𝑋 ∧ 𝑃) ∈ 𝐴)) |
8 | 7 | orim1d 965 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (((𝑋 ∧ 𝑃) = 𝑃 ∨ (𝑋 ∧ 𝑃) = 0 ) → ((𝑋 ∧ 𝑃) ∈ 𝐴 ∨ (𝑋 ∧ 𝑃) = 0 ))) |
9 | 5, 8 | mpd 15 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) ∈ 𝐴 ∨ (𝑋 ∧ 𝑃) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 meetcmee 18261 0.cp0 18372 OLcol 37982 Atomscatm 38071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-lat 18381 df-oposet 37984 df-ol 37986 df-covers 38074 df-ats 38075 |
This theorem is referenced by: 2at0mat0 38334 atmod1i1m 38667 |
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