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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 2622 | . . 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 35043 | . 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)𝑋}) |
6 | ssrab2 3687 | . 2 ⊢ {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾)𝑋} ⊆ 𝐴 | |
7 | 5, 6 | syl6eqss 3655 | 1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 Basecbs 15857 lecple 15948 Atomscatm 34550 pmapcpmap 34783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-pmap 34790 |
This theorem is referenced by: pmapssbaN 35046 pmapglb2N 35057 pmapglb2xN 35058 pmapjoin 35138 pmapjat1 35139 pmapjat2 35140 pmapjlln1 35141 hlmod1i 35142 polpmapN 35198 2pmaplubN 35212 pmapj2N 35215 pmapocjN 35216 polatN 35217 pmapsubclN 35232 ispsubcl2N 35233 pl42lem2N 35266 pl42lem3N 35267 |
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