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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 36922 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 𝑃 ∨ (𝑋 ∧ 𝑃) = 0 )) |
6 | eleq1a 2828 | . . . 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 2113 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 meetcmee 17664 0.cp0 17756 OLcol 36800 Atomscatm 36889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 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 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-proset 17647 df-poset 17665 df-plt 17677 df-lub 17693 df-glb 17694 df-join 17695 df-meet 17696 df-p0 17758 df-lat 17765 df-oposet 36802 df-ol 36804 df-covers 36892 df-ats 36893 |
This theorem is referenced by: 2at0mat0 37151 atmod1i1m 37484 |
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