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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 37310 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 𝑃 ∨ (𝑋 ∧ 𝑃) = 0 )) |
6 | eleq1a 2834 | . . . 4 ⊢ (𝑃 ∈ 𝐴 → ((𝑋 ∧ 𝑃) = 𝑃 → (𝑋 ∧ 𝑃) ∈ 𝐴)) | |
7 | 6 | 3ad2ant3 1134 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 𝑃 → (𝑋 ∧ 𝑃) ∈ 𝐴)) |
8 | 7 | orim1d 963 | . 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 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 meetcmee 18030 0.cp0 18141 OLcol 37188 Atomscatm 37277 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-lat 18150 df-oposet 37190 df-ol 37192 df-covers 37280 df-ats 37281 |
This theorem is referenced by: 2at0mat0 37539 atmod1i1m 37872 |
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