Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meetat | Structured version Visualization version GIF version |
Description: The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-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 |
---|---|
meetat | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 𝑃 ∨ (𝑋 ∧ 𝑃) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ollat 36353 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
2 | 1 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ Lat) |
3 | simp2 1133 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
4 | simp3 1134 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐴) | |
5 | m.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
6 | m.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | 5, 6 | atbase 36429 | . . . 4 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
8 | 4, 7 | syl 17 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐵) |
9 | eqid 2824 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | m.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
11 | 5, 9, 10 | latmle2 17690 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑋 ∧ 𝑃)(le‘𝐾)𝑃) |
12 | 2, 3, 8, 11 | syl3anc 1367 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ∧ 𝑃)(le‘𝐾)𝑃) |
13 | olop 36354 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
14 | 13 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ OP) |
15 | 5, 10 | latmcl 17665 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑋 ∧ 𝑃) ∈ 𝐵) |
16 | 2, 3, 8, 15 | syl3anc 1367 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ∧ 𝑃) ∈ 𝐵) |
17 | m.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
18 | 5, 9, 17, 6 | leatb 36432 | . . 3 ⊢ ((𝐾 ∈ OP ∧ (𝑋 ∧ 𝑃) ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃)(le‘𝐾)𝑃 ↔ ((𝑋 ∧ 𝑃) = 𝑃 ∨ (𝑋 ∧ 𝑃) = 0 ))) |
19 | 14, 16, 4, 18 | syl3anc 1367 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃)(le‘𝐾)𝑃 ↔ ((𝑋 ∧ 𝑃) = 𝑃 ∨ (𝑋 ∧ 𝑃) = 0 ))) |
20 | 12, 19 | mpbid 234 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 𝑃 ∨ (𝑋 ∧ 𝑃) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 lecple 16575 meetcmee 17558 0.cp0 17650 Latclat 17658 OPcops 36312 OLcol 36314 Atomscatm 36403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-lat 17659 df-oposet 36316 df-ol 36318 df-covers 36406 df-ats 36407 |
This theorem is referenced by: meetat2 36437 |
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