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Theorem pmapsub 35372
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 2651 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2651 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
4 pmapsub.m . . 3 𝑀 = (pmap‘𝐾)
51, 2, 3, 4pmapval 35361 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})
6 breq1 4688 . . . . . . . . . . . . . 14 (𝑐 = 𝑝 → (𝑐(le‘𝐾)𝑋𝑝(le‘𝐾)𝑋))
76elrab 3396 . . . . . . . . . . . . 13 (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋))
81, 3atbase 34894 . . . . . . . . . . . . . 14 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
98anim1i 591 . . . . . . . . . . . . 13 ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → (𝑝𝐵𝑝(le‘𝐾)𝑋))
107, 9sylbi 207 . . . . . . . . . . . 12 (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑝𝐵𝑝(le‘𝐾)𝑋))
11 breq1 4688 . . . . . . . . . . . . . 14 (𝑐 = 𝑞 → (𝑐(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))
1211elrab 3396 . . . . . . . . . . . . 13 (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋))
131, 3atbase 34894 . . . . . . . . . . . . . 14 (𝑞 ∈ (Atoms‘𝐾) → 𝑞𝐵)
1413anim1i 591 . . . . . . . . . . . . 13 ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋) → (𝑞𝐵𝑞(le‘𝐾)𝑋))
1512, 14sylbi 207 . . . . . . . . . . . 12 (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑞𝐵𝑞(le‘𝐾)𝑋))
1610, 15anim12i 589 . . . . . . . . . . 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 882 . . . . . . . . . . 11 (((𝑝𝐵𝑝(le‘𝐾)𝑋) ∧ (𝑞𝐵𝑞(le‘𝐾)𝑋)) ↔ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋)))
1816, 17sylib 208 . . . . . . . . . 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 592 . . . . . . . . 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 34894 . . . . . . . . 9 (𝑟 ∈ (Atoms‘𝐾) → 𝑟𝐵)
21 eqid 2651 . . . . . . . . . . . . . . . . 17 (join‘𝐾) = (join‘𝐾)
221, 2, 21latjle12 17109 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑞𝐵𝑋𝐵)) → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) ↔ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))
2322biimpd 219 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑞𝐵𝑋𝐵)) → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))
24233exp2 1307 . . . . . . . . . . . . . 14 (𝐾 ∈ Lat → (𝑝𝐵 → (𝑞𝐵 → (𝑋𝐵 → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)))))
2524impd 446 . . . . . . . . . . . . 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 620 . . . . . . . . . . 11 (((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)
2827adantr 480 . . . . . . . . . 10 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))) ∧ 𝑟𝐵) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)
291, 21latjcl 17098 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑝𝐵𝑞𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
30293expib 1287 . . . . . . . . . . . . 13 (𝐾 ∈ Lat → ((𝑝𝐵𝑞𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵))
311, 2lattr 17103 . . . . . . . . . . . . . . 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 1307 . . . . . . . . . . . . . 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 618 . . . . . . . . . . 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 757 . . . . . . . . . 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 710 . . . . . . . . 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 493 . . . . . . . 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 476 . . . . . . . 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 565 . . . . . . 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 4688 . . . . . . . 8 (𝑐 = 𝑟 → (𝑐(le‘𝐾)𝑋𝑟(le‘𝐾)𝑋))
4241elrab 3396 . . . . . . 7 (𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑋))
4340, 42syl6ibr 242 . . . . . 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 2995 . . . . 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 3000 . . . 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 3720 . . . 4 {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾)
4745, 46jctil 559 . . 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 35349 . . . 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 480 . . 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 247 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∈ 𝑆)
525, 51eqeltrd 2730 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  wss 3607   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  Latclat 17092  Atomscatm 34868  PSubSpcpsubsp 35100  pmapcpmap 35101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-poset 16993  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-lat 17093  df-ats 34872  df-psubsp 35107  df-pmap 35108
This theorem is referenced by:  hlmod1i  35460  polsubN  35511  pl42lem4N  35586
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