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Theorem pmapsub 35567
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 2817 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2817 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
4 pmapsub.m . . 3 𝑀 = (pmap‘𝐾)
51, 2, 3, 4pmapval 35556 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})
6 breq1 4858 . . . . . . . . . . . . . 14 (𝑐 = 𝑝 → (𝑐(le‘𝐾)𝑋𝑝(le‘𝐾)𝑋))
76elrab 3570 . . . . . . . . . . . . 13 (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋))
81, 3atbase 35088 . . . . . . . . . . . . . 14 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
98anim1i 604 . . . . . . . . . . . . 13 ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → (𝑝𝐵𝑝(le‘𝐾)𝑋))
107, 9sylbi 208 . . . . . . . . . . . 12 (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑝𝐵𝑝(le‘𝐾)𝑋))
11 breq1 4858 . . . . . . . . . . . . . 14 (𝑐 = 𝑞 → (𝑐(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))
1211elrab 3570 . . . . . . . . . . . . 13 (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋))
131, 3atbase 35088 . . . . . . . . . . . . . 14 (𝑞 ∈ (Atoms‘𝐾) → 𝑞𝐵)
1413anim1i 604 . . . . . . . . . . . . 13 ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋) → (𝑞𝐵𝑞(le‘𝐾)𝑋))
1512, 14sylbi 208 . . . . . . . . . . . 12 (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑞𝐵𝑞(le‘𝐾)𝑋))
1610, 15anim12i 602 . . . . . . . . . . 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 638 . . . . . . . . . . 11 (((𝑝𝐵𝑝(le‘𝐾)𝑋) ∧ (𝑞𝐵𝑞(le‘𝐾)𝑋)) ↔ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋)))
1816, 17sylib 209 . . . . . . . . . 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 605 . . . . . . . . 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 35088 . . . . . . . . 9 (𝑟 ∈ (Atoms‘𝐾) → 𝑟𝐵)
21 eqid 2817 . . . . . . . . . . . . . . . . 17 (join‘𝐾) = (join‘𝐾)
221, 2, 21latjle12 17287 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑞𝐵𝑋𝐵)) → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) ↔ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))
2322biimpd 220 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑞𝐵𝑋𝐵)) → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))
24233exp2 1456 . . . . . . . . . . . . . 14 (𝐾 ∈ Lat → (𝑝𝐵 → (𝑞𝐵 → (𝑋𝐵 → ((𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)))))
2524impd 398 . . . . . . . . . . . . 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 416 . . . . . . . . . . 11 (((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)
2827adantr 468 . . . . . . . . . 10 ((((𝐾 ∈ Lat ∧ 𝑋𝐵) ∧ ((𝑝𝐵𝑞𝐵) ∧ (𝑝(le‘𝐾)𝑋𝑞(le‘𝐾)𝑋))) ∧ 𝑟𝐵) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)
291, 21latjcl 17276 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑝𝐵𝑞𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
30293expib 1145 . . . . . . . . . . . . 13 (𝐾 ∈ Lat → ((𝑝𝐵𝑞𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵))
311, 2lattr 17281 . . . . . . . . . . . . . . 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 1456 . . . . . . . . . . . . . 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 414 . . . . . . . . . . 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 703 . . . . . . . . . 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 677 . . . . . . . . 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 585 . . . . . . . 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 473 . . . . . . . 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 517 . . . . . . 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 4858 . . . . . . . 8 (𝑐 = 𝑟 → (𝑐(le‘𝐾)𝑋𝑟(le‘𝐾)𝑋))
4241elrab 3570 . . . . . . 7 (𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑋))
4340, 42syl6ibr 243 . . . . . 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 3165 . . . . 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 3170 . . . 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 3895 . . . 4 {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾)
4745, 46jctil 511 . . 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 35544 . . . 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 468 . . 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 248 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∈ 𝑆)
525, 51eqeltrd 2896 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  wral 3107  {crab 3111  wss 3780   class class class wbr 4855  cfv 6111  (class class class)co 6884  Basecbs 16088  lecple 16180  joincjn 17169  Latclat 17270  Atomscatm 35062  PSubSpcpsubsp 35295  pmapcpmap 35296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-riota 6845  df-ov 6887  df-oprab 6888  df-poset 17171  df-lub 17199  df-glb 17200  df-join 17201  df-meet 17202  df-lat 17271  df-ats 35066  df-psubsp 35302  df-pmap 35303
This theorem is referenced by:  hlmod1i  35655  polsubN  35706  pl42lem4N  35781
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