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 2821 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | pmap1.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
3 | 1, 2 | op1cl 36336 | . . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
4 | eqid 2821 | . . . 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 36908 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 }) |
8 | 3, 7 | mpdan 685 | . 2 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 }) |
9 | 1, 5 | atbase 36440 | . . . . 5 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾)) |
10 | 1, 4, 2 | ople1 36342 | . . . . 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 3182 | . . 3 ⊢ (𝐾 ∈ OP → ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 ) |
13 | rabid2 3381 | . . 3 ⊢ (𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 } ↔ ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 ) | |
14 | 12, 13 | sylibr 236 | . 2 ⊢ (𝐾 ∈ OP → 𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 }) |
15 | 8, 14 | eqtr4d 2859 | 1 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 class class class wbr 5066 ‘cfv 6355 Basecbs 16483 lecple 16572 1.cp1 17648 OPcops 36323 Atomscatm 36414 pmapcpmap 36648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-lub 17584 df-p1 17650 df-oposet 36327 df-ats 36418 df-pmap 36655 |
This theorem is referenced by: pmapglb2N 36922 pmapglb2xN 36923 |
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