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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 2734 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | pmap1.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
3 | 1, 2 | op1cl 39090 | . . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
4 | eqid 2734 | . . . 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 39663 | . . 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 39194 | . . . . 5 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾)) |
10 | 1, 4, 2 | ople1 39096 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾) 1 ) |
11 | 9, 10 | sylan2 592 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐴) → 𝑝(le‘𝐾) 1 ) |
12 | 11 | ralrimiva 3148 | . . 3 ⊢ (𝐾 ∈ OP → ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 ) |
13 | rabid2 3472 | . . 3 ⊢ (𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 } ↔ ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 ) | |
14 | 12, 13 | sylibr 234 | . 2 ⊢ (𝐾 ∈ OP → 𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 }) |
15 | 8, 14 | eqtr4d 2777 | 1 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 ∀wral 3063 {crab 3438 class class class wbr 5169 ‘cfv 6572 Basecbs 17253 lecple 17313 1.cp1 18489 OPcops 39077 Atomscatm 39168 pmapcpmap 39403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-lub 18411 df-p1 18491 df-oposet 39081 df-ats 39172 df-pmap 39410 |
This theorem is referenced by: pmapglb2N 39677 pmapglb2xN 39678 |
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