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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 2733 | . . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
2 | pmapmeet.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
3 | simp1 1137 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ HL) | |
4 | simp2 1138 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
5 | simp3 1139 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | meetval 18340 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌})) |
7 | 6 | fveq2d 6892 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃‘(𝑋 ∧ 𝑌)) = (𝑃‘((glb‘𝐾)‘{𝑋, 𝑌}))) |
8 | prssi 4823 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → {𝑋, 𝑌} ⊆ 𝐵) | |
9 | 8 | 3adant1 1131 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → {𝑋, 𝑌} ⊆ 𝐵) |
10 | prnzg 4781 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → {𝑋, 𝑌} ≠ ∅) | |
11 | 10 | 3ad2ant2 1135 | . . 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 38579 | . . 3 ⊢ ((𝐾 ∈ HL ∧ {𝑋, 𝑌} ⊆ 𝐵 ∧ {𝑋, 𝑌} ≠ ∅) → (𝑃‘((glb‘𝐾)‘{𝑋, 𝑌})) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝑃‘𝑥)) |
15 | 3, 9, 11, 14 | syl3anc 1372 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃‘((glb‘𝐾)‘{𝑋, 𝑌})) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝑃‘𝑥)) |
16 | fveq2 6888 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑃‘𝑥) = (𝑃‘𝑋)) | |
17 | fveq2 6888 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝑃‘𝑥) = (𝑃‘𝑌)) | |
18 | 16, 17 | iinxprg 5091 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝑃‘𝑥) = ((𝑃‘𝑋) ∩ (𝑃‘𝑌))) |
19 | 18 | 3adant1 1131 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝑃‘𝑥) = ((𝑃‘𝑋) ∩ (𝑃‘𝑌))) |
20 | 7, 15, 19 | 3eqtrd 2777 | 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 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 {cpr 4629 ∩ ciin 4997 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 glbcglb 18259 meetcmee 18261 Atomscatm 38071 HLchlt 38158 pmapcpmap 38306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-poset 18262 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-lat 18381 df-clat 18448 df-ats 38075 df-hlat 38159 df-pmap 38313 |
This theorem is referenced by: hlmod1i 38665 poldmj1N 38737 pmapj2N 38738 pnonsingN 38742 psubclinN 38757 poml4N 38762 pl42lem1N 38788 pl42lem2N 38789 |
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