| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > pmap0 | Structured version Visualization version GIF version | ||
| Description: Value of the projective map of a Hilbert lattice at lattice zero. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.) |
| Ref | Expression |
|---|---|
| pmap0.z | ⊢ 0 = (0.‘𝐾) |
| pmap0.m | ⊢ 𝑀 = (pmap‘𝐾) |
| Ref | Expression |
|---|---|
| pmap0 | ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 2 | pmap0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
| 3 | 1, 2 | atl0cl 39304 | . . 3 ⊢ (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾)) |
| 4 | eqid 2737 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 5 | eqid 2737 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
| 6 | pmap0.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
| 7 | 1, 4, 5, 6 | pmapval 39759 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 0 ∈ (Base‘𝐾)) → (𝑀‘ 0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 }) |
| 8 | 3, 7 | mpdan 687 | . 2 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 }) |
| 9 | 4, 2, 5 | atnle0 39310 | . . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑎 ∈ (Atoms‘𝐾)) → ¬ 𝑎(le‘𝐾) 0 ) |
| 10 | 9 | nrexdv 3149 | . . . 4 ⊢ (𝐾 ∈ AtLat → ¬ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 ) |
| 11 | rabn0 4389 | . . . 4 ⊢ ({𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 ) | |
| 12 | 10, 11 | sylnibr 329 | . . 3 ⊢ (𝐾 ∈ AtLat → ¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅) |
| 13 | nne 2944 | . . 3 ⊢ (¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅) | |
| 14 | 12, 13 | sylib 218 | . 2 ⊢ (𝐾 ∈ AtLat → {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅) |
| 15 | 8, 14 | eqtrd 2777 | 1 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 {crab 3436 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 0.cp0 18468 Atomscatm 39264 AtLatcal 39265 pmapcpmap 39499 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-proset 18340 df-poset 18359 df-plt 18375 df-glb 18392 df-p0 18470 df-lat 18477 df-covers 39267 df-ats 39268 df-atl 39299 df-pmap 39506 |
| This theorem is referenced by: pmapeq0 39768 pmapjat1 39855 pol1N 39912 pnonsingN 39935 |
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