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 unit. 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 2737 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | pmap1.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
3 | 1, 2 | op1cl 36936 | . . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
4 | eqid 2737 | . . . 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 37508 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 }) |
8 | 3, 7 | mpdan 687 | . 2 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 }) |
9 | 1, 5 | atbase 37040 | . . . . 5 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾)) |
10 | 1, 4, 2 | ople1 36942 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾) 1 ) |
11 | 9, 10 | sylan2 596 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐴) → 𝑝(le‘𝐾) 1 ) |
12 | 11 | ralrimiva 3105 | . . 3 ⊢ (𝐾 ∈ OP → ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 ) |
13 | rabid2 3293 | . . 3 ⊢ (𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 } ↔ ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 ) | |
14 | 12, 13 | sylibr 237 | . 2 ⊢ (𝐾 ∈ OP → 𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 }) |
15 | 8, 14 | eqtr4d 2780 | 1 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ∀wral 3061 {crab 3065 class class class wbr 5053 ‘cfv 6380 Basecbs 16760 lecple 16809 1.cp1 17930 OPcops 36923 Atomscatm 37014 pmapcpmap 37248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-lub 17852 df-p1 17932 df-oposet 36927 df-ats 37018 df-pmap 37255 |
This theorem is referenced by: pmapglb2N 37522 pmapglb2xN 37523 |
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