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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 2821 | . . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
2 | pmapmeet.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
3 | simp1 1132 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ HL) | |
4 | simp2 1133 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
5 | simp3 1134 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | meetval 17629 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌})) |
7 | 6 | fveq2d 6674 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃‘(𝑋 ∧ 𝑌)) = (𝑃‘((glb‘𝐾)‘{𝑋, 𝑌}))) |
8 | prssi 4754 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → {𝑋, 𝑌} ⊆ 𝐵) | |
9 | 8 | 3adant1 1126 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → {𝑋, 𝑌} ⊆ 𝐵) |
10 | prnzg 4713 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → {𝑋, 𝑌} ≠ ∅) | |
11 | 10 | 3ad2ant2 1130 | . . 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 36921 | . . 3 ⊢ ((𝐾 ∈ HL ∧ {𝑋, 𝑌} ⊆ 𝐵 ∧ {𝑋, 𝑌} ≠ ∅) → (𝑃‘((glb‘𝐾)‘{𝑋, 𝑌})) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝑃‘𝑥)) |
15 | 3, 9, 11, 14 | syl3anc 1367 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃‘((glb‘𝐾)‘{𝑋, 𝑌})) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝑃‘𝑥)) |
16 | fveq2 6670 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑃‘𝑥) = (𝑃‘𝑋)) | |
17 | fveq2 6670 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝑃‘𝑥) = (𝑃‘𝑌)) | |
18 | 16, 17 | iinxprg 5011 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝑃‘𝑥) = ((𝑃‘𝑋) ∩ (𝑃‘𝑌))) |
19 | 18 | 3adant1 1126 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝑃‘𝑥) = ((𝑃‘𝑋) ∩ (𝑃‘𝑌))) |
20 | 7, 15, 19 | 3eqtrd 2860 | 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 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {cpr 4569 ∩ ciin 4920 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 glbcglb 17553 meetcmee 17555 Atomscatm 36414 HLchlt 36501 pmapcpmap 36648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-poset 17556 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-lat 17656 df-clat 17718 df-ats 36418 df-hlat 36502 df-pmap 36655 |
This theorem is referenced by: hlmod1i 37007 poldmj1N 37079 pmapj2N 37080 pnonsingN 37084 psubclinN 37099 poml4N 37104 pl42lem1N 37130 pl42lem2N 37131 |
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