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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapmeet | Structured version Visualization version GIF version |
Description: The projective map of a meet. (Contributed by NM, 25-Jan-2012.) |
Ref | Expression |
---|---|
pmapmeet.b | ⊢ 𝐵 = (Base‘𝐾) |
pmapmeet.m | ⊢ ∧ = (meet‘𝐾) |
pmapmeet.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pmapmeet.p | ⊢ 𝑃 = (pmap‘𝐾) |
Ref | Expression |
---|---|
pmapmeet | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃‘(𝑋 ∧ 𝑌)) = ((𝑃‘𝑋) ∩ (𝑃‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
2 | pmapmeet.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
3 | simp1 1054 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ HL) | |
4 | simp2 1055 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
5 | simp3 1056 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | meetval 16842 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌})) |
7 | 6 | fveq2d 6107 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃‘(𝑋 ∧ 𝑌)) = (𝑃‘((glb‘𝐾)‘{𝑋, 𝑌}))) |
8 | prssi 4293 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → {𝑋, 𝑌} ⊆ 𝐵) | |
9 | 8 | 3adant1 1072 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → {𝑋, 𝑌} ⊆ 𝐵) |
10 | prnzg 4254 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → {𝑋, 𝑌} ≠ ∅) | |
11 | 10 | 3ad2ant2 1076 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → {𝑋, 𝑌} ≠ ∅) |
12 | pmapmeet.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
13 | pmapmeet.p | . . . 4 ⊢ 𝑃 = (pmap‘𝐾) | |
14 | 12, 1, 13 | pmapglb 34074 | . . 3 ⊢ ((𝐾 ∈ HL ∧ {𝑋, 𝑌} ⊆ 𝐵 ∧ {𝑋, 𝑌} ≠ ∅) → (𝑃‘((glb‘𝐾)‘{𝑋, 𝑌})) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝑃‘𝑥)) |
15 | 3, 9, 11, 14 | syl3anc 1318 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃‘((glb‘𝐾)‘{𝑋, 𝑌})) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝑃‘𝑥)) |
16 | fveq2 6103 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑃‘𝑥) = (𝑃‘𝑋)) | |
17 | fveq2 6103 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝑃‘𝑥) = (𝑃‘𝑌)) | |
18 | 16, 17 | iinxprg 4537 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝑃‘𝑥) = ((𝑃‘𝑋) ∩ (𝑃‘𝑌))) |
19 | 18 | 3adant1 1072 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝑃‘𝑥) = ((𝑃‘𝑋) ∩ (𝑃‘𝑌))) |
20 | 7, 15, 19 | 3eqtrd 2648 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃‘(𝑋 ∧ 𝑌)) = ((𝑃‘𝑋) ∩ (𝑃‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {cpr 4127 ∩ ciin 4456 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 glbcglb 16766 meetcmee 16768 Atomscatm 33568 HLchlt 33655 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-8 1979 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-pow 4769 ax-pr 4833 ax-un 6847 |
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-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 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-riota 6511 df-ov 6552 df-oprab 6553 df-poset 16769 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-lat 16869 df-clat 16931 df-ats 33572 df-hlat 33656 df-pmap 33808 |
This theorem is referenced by: hlmod1i 34160 poldmj1N 34232 pmapj2N 34233 pnonsingN 34237 psubclinN 34252 poml4N 34257 pl42lem1N 34283 pl42lem2N 34284 |
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