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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 3185 | . 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V) | |
2 | pmapfval.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
3 | fveq2 6103 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
4 | pmapfval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | 3, 4 | syl6eqr 2662 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
6 | fveq2 6103 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾)) | |
7 | pmapfval.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | 6, 7 | syl6eqr 2662 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴) |
9 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾)) | |
10 | pmapfval.l | . . . . . . . 8 ⊢ ≤ = (le‘𝐾) | |
11 | 9, 10 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ ) |
12 | 11 | breqd 4594 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑎(le‘𝑘)𝑥 ↔ 𝑎 ≤ 𝑥)) |
13 | 8, 12 | rabeqbidv 3168 | . . . . 5 ⊢ (𝑘 = 𝐾 → {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥} = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}) |
14 | 5, 13 | mpteq12dv 4663 | . . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) |
15 | df-pmap 33808 | . . . 4 ⊢ pmap = (𝑘 ∈ V ↦ (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥})) | |
16 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝐾) ∈ V | |
17 | 4, 16 | eqeltri 2684 | . . . . 5 ⊢ 𝐵 ∈ V |
18 | 17 | mptex 6390 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}) ∈ V |
19 | 14, 15, 18 | fvmpt 6191 | . . 3 ⊢ (𝐾 ∈ V → (pmap‘𝐾) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) |
20 | 2, 19 | syl5eq 2656 | . 2 ⊢ (𝐾 ∈ V → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) |
21 | 1, 20 | syl 17 | 1 ⊢ (𝐾 ∈ 𝐶 → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 Basecbs 15695 lecple 15775 Atomscatm 33568 pmapcpmap 33801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-pmap 33808 |
This theorem is referenced by: pmapval 34061 |
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