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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapfval | Structured version Visualization version GIF version |
Description: The projective map of a Hilbert lattice. (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 |
---|---|
pmapfval | ⊢ (𝐾 ∈ 𝐶 → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3439 | . 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V) | |
2 | pmapfval.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
3 | fveq2 6736 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
4 | pmapfval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | 3, 4 | eqtr4di 2797 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
6 | fveq2 6736 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾)) | |
7 | pmapfval.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | 6, 7 | eqtr4di 2797 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴) |
9 | fveq2 6736 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾)) | |
10 | pmapfval.l | . . . . . . . 8 ⊢ ≤ = (le‘𝐾) | |
11 | 9, 10 | eqtr4di 2797 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ ) |
12 | 11 | breqd 5079 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑎(le‘𝑘)𝑥 ↔ 𝑎 ≤ 𝑥)) |
13 | 8, 12 | rabeqbidv 3409 | . . . . 5 ⊢ (𝑘 = 𝐾 → {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥} = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}) |
14 | 5, 13 | mpteq12dv 5155 | . . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) |
15 | df-pmap 37285 | . . . 4 ⊢ pmap = (𝑘 ∈ V ↦ (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥})) | |
16 | 14, 15, 4 | mptfvmpt 7063 | . . 3 ⊢ (𝐾 ∈ V → (pmap‘𝐾) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) |
17 | 2, 16 | syl5eq 2791 | . 2 ⊢ (𝐾 ∈ V → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) |
18 | 1, 17 | syl 17 | 1 ⊢ (𝐾 ∈ 𝐶 → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2111 {crab 3066 Vcvv 3421 class class class wbr 5068 ↦ cmpt 5150 ‘cfv 6398 Basecbs 16788 lecple 16837 Atomscatm 37044 pmapcpmap 37278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pr 5337 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-pmap 37285 |
This theorem is referenced by: pmapval 37538 |
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