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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmap1N | Structured version Visualization version GIF version |
Description: Value of the projective map of a Hilbert lattice at lattice unity. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pmap1.u | ⊢ 1 = (1.‘𝐾) |
pmap1.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pmap1.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
pmap1N | ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | pmap1.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
3 | 1, 2 | op1cl 37992 | . . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
4 | eqid 2733 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | pmap1.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | pmap1.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
7 | 1, 4, 5, 6 | pmapval 38565 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 }) |
8 | 3, 7 | mpdan 686 | . 2 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 }) |
9 | 1, 5 | atbase 38096 | . . . . 5 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾)) |
10 | 1, 4, 2 | ople1 37998 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾) 1 ) |
11 | 9, 10 | sylan2 594 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐴) → 𝑝(le‘𝐾) 1 ) |
12 | 11 | ralrimiva 3147 | . . 3 ⊢ (𝐾 ∈ OP → ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 ) |
13 | rabid2 3465 | . . 3 ⊢ (𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 } ↔ ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 ) | |
14 | 12, 13 | sylibr 233 | . 2 ⊢ (𝐾 ∈ OP → 𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 }) |
15 | 8, 14 | eqtr4d 2776 | 1 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 class class class wbr 5146 ‘cfv 6539 Basecbs 17139 lecple 17199 1.cp1 18372 OPcops 37979 Atomscatm 38070 pmapcpmap 38305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-lub 18294 df-p1 18374 df-oposet 37983 df-ats 38074 df-pmap 38312 |
This theorem is referenced by: pmapglb2N 38579 pmapglb2xN 38580 |
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