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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapval | Structured version Visualization version GIF version |
Description: Value of the projective map of a Hilbert lattice. Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.) |
Ref | Expression |
---|---|
pmapfval.b | ⊢ 𝐵 = (Base‘𝐾) |
pmapfval.l | ⊢ ≤ = (le‘𝐾) |
pmapfval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pmapfval.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
pmapval | ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmapfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | pmapfval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | pmapfval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | pmapfval.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
5 | 1, 2, 3, 4 | pmapfval 39739 | . . 3 ⊢ (𝐾 ∈ 𝐶 → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) |
6 | 5 | fveq1d 6909 | . 2 ⊢ (𝐾 ∈ 𝐶 → (𝑀‘𝑋) = ((𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})‘𝑋)) |
7 | breq2 5152 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑎 ≤ 𝑥 ↔ 𝑎 ≤ 𝑋)) | |
8 | 7 | rabbidv 3441 | . . 3 ⊢ (𝑥 = 𝑋 → {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥} = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋}) |
9 | eqid 2735 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}) | |
10 | 3 | fvexi 6921 | . . . 4 ⊢ 𝐴 ∈ V |
11 | 10 | rabex 5345 | . . 3 ⊢ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋} ∈ V |
12 | 8, 9, 11 | fvmpt 7016 | . 2 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋}) |
13 | 6, 12 | sylan9eq 2795 | 1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 Basecbs 17245 lecple 17305 Atomscatm 39245 pmapcpmap 39480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-pmap 39487 |
This theorem is referenced by: elpmap 39741 pmapssat 39742 pmaple 39744 pmapat 39746 pmap0 39748 pmap1N 39750 pmapsub 39751 pmapglbx 39752 isline2 39757 linepmap 39758 polpmapN 39895 2polssN 39898 pmaplubN 39907 |
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