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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 2734 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | pmap0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
3 | 1, 2 | atl0cl 39284 | . . 3 ⊢ (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾)) |
4 | eqid 2734 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | eqid 2734 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | pmap0.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
7 | 1, 4, 5, 6 | pmapval 39739 | . . 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 39290 | . . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑎 ∈ (Atoms‘𝐾)) → ¬ 𝑎(le‘𝐾) 0 ) |
10 | 9 | nrexdv 3146 | . . . 4 ⊢ (𝐾 ∈ AtLat → ¬ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 ) |
11 | rabn0 4394 | . . . 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 2941 | . . 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 2774 | 1 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 {crab 3432 ∅c0 4338 class class class wbr 5147 ‘cfv 6562 Basecbs 17244 lecple 17304 0.cp0 18480 Atomscatm 39244 AtLatcal 39245 pmapcpmap 39479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-proset 18351 df-poset 18370 df-plt 18387 df-glb 18404 df-p0 18482 df-lat 18489 df-covers 39247 df-ats 39248 df-atl 39279 df-pmap 39486 |
This theorem is referenced by: pmapeq0 39748 pmapjat1 39835 pol1N 39892 pnonsingN 39915 |
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