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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 37052 | . . 3 ⊢ (𝐾 ∈ 𝐶 → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) |
6 | 5 | fveq1d 6647 | . 2 ⊢ (𝐾 ∈ 𝐶 → (𝑀‘𝑋) = ((𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})‘𝑋)) |
7 | breq2 5034 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑎 ≤ 𝑥 ↔ 𝑎 ≤ 𝑋)) | |
8 | 7 | rabbidv 3427 | . . 3 ⊢ (𝑥 = 𝑋 → {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥} = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋}) |
9 | eqid 2798 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}) | |
10 | 3 | fvexi 6659 | . . . 4 ⊢ 𝐴 ∈ V |
11 | 10 | rabex 5199 | . . 3 ⊢ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋} ∈ V |
12 | 8, 9, 11 | fvmpt 6745 | . 2 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋}) |
13 | 6, 12 | sylan9eq 2853 | 1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 Basecbs 16475 lecple 16564 Atomscatm 36559 pmapcpmap 36793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-pmap 36800 |
This theorem is referenced by: elpmap 37054 pmapssat 37055 pmaple 37057 pmapat 37059 pmap0 37061 pmap1N 37063 pmapsub 37064 pmapglbx 37065 isline2 37070 linepmap 37071 polpmapN 37208 2polssN 37211 pmaplubN 37220 |
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