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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 3500 | . 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V) | |
| 2 | pmapfval.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
| 3 | fveq2 6905 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) | |
| 4 | pmapfval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
| 5 | 3, 4 | eqtr4di 2794 | . . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) | 
| 6 | fveq2 6905 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾)) | |
| 7 | pmapfval.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 8 | 6, 7 | eqtr4di 2794 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴) | 
| 9 | fveq2 6905 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾)) | |
| 10 | pmapfval.l | . . . . . . . 8 ⊢ ≤ = (le‘𝐾) | |
| 11 | 9, 10 | eqtr4di 2794 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ ) | 
| 12 | 11 | breqd 5153 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑎(le‘𝑘)𝑥 ↔ 𝑎 ≤ 𝑥)) | 
| 13 | 8, 12 | rabeqbidv 3454 | . . . . 5 ⊢ (𝑘 = 𝐾 → {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥} = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}) | 
| 14 | 5, 13 | mpteq12dv 5232 | . . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) | 
| 15 | df-pmap 39507 | . . . 4 ⊢ pmap = (𝑘 ∈ V ↦ (𝑥 ∈ (Base‘𝑘) ↦ {𝑎 ∈ (Atoms‘𝑘) ∣ 𝑎(le‘𝑘)𝑥})) | |
| 16 | 14, 15, 4 | mptfvmpt 7249 | . . 3 ⊢ (𝐾 ∈ V → (pmap‘𝐾) = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥})) | 
| 17 | 2, 16 | eqtrid 2788 | . 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 1539 ∈ wcel 2107 {crab 3435 Vcvv 3479 class class class wbr 5142 ↦ cmpt 5224 ‘cfv 6560 Basecbs 17248 lecple 17305 Atomscatm 39265 pmapcpmap 39500 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-pmap 39507 | 
| This theorem is referenced by: pmapval 39760 | 
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