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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapssat | Structured version Visualization version GIF version |
Description: The projective map of a Hilbert lattice is a set of atoms. (Contributed by NM, 14-Jan-2012.) |
Ref | Expression |
---|---|
pmapssat.b | ⊢ 𝐵 = (Base‘𝐾) |
pmapssat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pmapssat.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
pmapssat | ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmapssat.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2610 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | pmapssat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | pmapssat.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
5 | 1, 2, 3, 4 | pmapval 34061 | . 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)𝑋}) |
6 | ssrab2 3650 | . 2 ⊢ {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)𝑋} ⊆ 𝐴 | |
7 | 5, 6 | syl6eqss 3618 | 1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ⊆ wss 3540 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 Atomscatm 33568 pmapcpmap 33801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-pmap 33808 |
This theorem is referenced by: pmapssbaN 34064 pmapglb2N 34075 pmapglb2xN 34076 pmapjoin 34156 pmapjat1 34157 pmapjat2 34158 pmapjlln1 34159 hlmod1i 34160 polpmapN 34216 2pmaplubN 34230 pmapj2N 34233 pmapocjN 34234 polatN 34235 pmapsubclN 34250 ispsubcl2N 34251 pl42lem2N 34284 pl42lem3N 34285 |
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