Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmapsub Structured version   Visualization version   GIF version

Theorem pmapsub 36784
Description: The projective map of a Hilbert lattice maps to projective subspaces. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)
Hypotheses
Ref Expression
pmapsub.b 𝐵 = (Base‘𝐾)
pmapsub.s 𝑆 = (PSubSp‘𝐾)
pmapsub.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
pmapsub ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝑆)

Proof of Theorem pmapsub
Dummy variables 𝑞 𝑝 𝑟 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pmapsub.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2818 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2818 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
4 pmapsub.m . . 3 𝑀 = (pmap‘𝐾)
51, 2, 3, 4pmapval 36773 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})
6 breq1 5060 . . . . . . . . . . . . . 14 (𝑐 = 𝑝 → (𝑐(le‘𝐾)𝑋𝑝(le‘𝐾)𝑋))
76elrab 3677 . . . . . . . . . . . . 13 (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋))
81, 3atbase 36305 . . . . . . . . . . . . . 14 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
98anim1i 614 . . . . . . . . . . . . 13 ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → (𝑝𝐵𝑝(le‘𝐾)𝑋))
107, 9sylbi 218 . . . . . . . . . . . 12 (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑝𝐵𝑝(le‘𝐾)𝑋))
11 breq1 5060 . . . . . . . . . . . . . 14 (𝑐 = 𝑞 → (𝑐(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))
1211elrab 3677 . . . . . . . . . . . . 13 (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋))
131, 3atbase 36305 . . . . . . . . . . . . . 14 (𝑞 ∈ (Atoms‘𝐾) → 𝑞𝐵)
1413anim1i 614 . . . . . . . . . . . . 13 ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋) → (𝑞𝐵𝑞(le‘𝐾)𝑋))
1512, 14sylbi 218 . . . . . . . . . . . 12 (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑞𝐵𝑞(le‘𝐾)𝑋))
1610, 15anim12i 612 . . . . . . . . . . 11 ((𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}) → ((𝑝𝐵𝑝(le‘𝐾)𝑋) ∧ (𝑞𝐵𝑞(le‘𝐾)𝑋)))
17 an4 652 . . . . . . . . . . 11 (((𝑝𝐵𝑝(le‘𝐾)𝑋) ∧ (𝑞𝐵𝑞(le‘𝐾)𝑋)) ↔ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋)))
1816, 17sylib 219 . . . . . . . . . 10 ((𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}) → ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋)))
1918anim2i 616 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) → ((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))))
201, 3atbase 36305 . . . . . . . . 9 (𝑟 ∈ (Atoms‘𝐾) → 𝑟𝐵)
21 eqid 2818 . . . . . . . . . . . . . . . . 17 (join‘𝐾) = (join‘𝐾)
221, 2, 21latjle12 17660 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑞𝐵𝑋𝐵)) → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) ↔ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))
2322biimpd 230 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑞𝐵𝑋𝐵)) → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))
24233exp2 1346 . . . . . . . . . . . . . 14 (𝐾 ∈ Lat → (𝑝𝐵 → (𝑞𝐵 → (𝑋𝐵 → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)))))
2524impd 411 . . . . . . . . . . . . 13 (𝐾 ∈ Lat → ((𝑝𝐵𝑞𝐵) → (𝑋𝐵 → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))))
2625com23 86 . . . . . . . . . . . 12 (𝐾 ∈ Lat → (𝑋𝐵 → ((𝑝𝐵𝑞𝐵) → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))))
2726imp43 428 . . . . . . . . . . 11 (((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)
2827adantr 481 . . . . . . . . . 10 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))) ∧ 𝑟𝐵) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)
291, 21latjcl 17649 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑝𝐵𝑞𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
30293expib 1114 . . . . . . . . . . . . 13 (𝐾 ∈ Lat → ((𝑝𝐵𝑞𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵))
311, 2lattr 17654 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑟𝐵 ∧ (𝑝(join‘𝐾)𝑞) ∈ 𝐵𝑋𝐵)) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋))
32313exp2 1346 . . . . . . . . . . . . . 14 (𝐾 ∈ Lat → (𝑟𝐵 → ((𝑝(join‘𝐾)𝑞) ∈ 𝐵 → (𝑋𝐵 → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋)))))
3332com24 95 . . . . . . . . . . . . 13 (𝐾 ∈ Lat → (𝑋𝐵 → ((𝑝(join‘𝐾)𝑞) ∈ 𝐵 → (𝑟𝐵 → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋)))))
3430, 33syl5d 73 . . . . . . . . . . . 12 (𝐾 ∈ Lat → (𝑋𝐵 → ((𝑝𝐵𝑞𝐵) → (𝑟𝐵 → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋)))))
3534imp41 426 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝𝐵𝑞𝐵)) ∧ 𝑟𝐵) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋))
3635adantlrr 717 . . . . . . . . . 10 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))) ∧ 𝑟𝐵) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋))
3728, 36mpan2d 690 . . . . . . . . 9 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))) ∧ 𝑟𝐵) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟(le‘𝐾)𝑋))
3819, 20, 37syl2an 595 . . . . . . . 8 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟(le‘𝐾)𝑋))
39 simpr 485 . . . . . . . 8 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) ∧ 𝑟 ∈ (Atoms‘𝐾)) → 𝑟 ∈ (Atoms‘𝐾))
4038, 39jctild 526 . . . . . . 7 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑋)))
41 breq1 5060 . . . . . . . 8 (𝑐 = 𝑟 → (𝑐(le‘𝐾)𝑋𝑟(le‘𝐾)𝑋))
4241elrab 3677 . . . . . . 7 (𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑋))
4340, 42syl6ibr 253 . . . . . 6 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))
4443ralrimiva 3179 . . . . 5 (((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) → ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))
4544ralrimivva 3188 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → ∀𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))
46 ssrab2 4053 . . . 4 {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾)
4745, 46jctil 520 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})))
48 pmapsub.s . . . . 5 𝑆 = (PSubSp‘𝐾)
492, 21, 3, 48ispsubsp 36761 . . . 4 (𝐾 ∈ Lat → ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∈ 𝑆 ↔ ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))))
5049adantr 481 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∈ 𝑆 ↔ ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))))
5147, 50mpbird 258 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∈ 𝑆)
525, 51eqeltrd 2910 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  {crab 3139  wss 3933   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  Latclat 17643  Atomscatm 36279  PSubSpcpsubsp 36512  pmapcpmap 36513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-poset 17544  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-lat 17644  df-ats 36283  df-psubsp 36519  df-pmap 36520
This theorem is referenced by:  hlmod1i  36872  polsubN  36923  pl42lem4N  36998
  Copyright terms: Public domain W3C validator