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Theorem psubspi 34513
 Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012.)
Hypotheses
Ref Expression
psubspset.l = (le‘𝐾)
psubspset.j = (join‘𝐾)
psubspset.a 𝐴 = (Atoms‘𝐾)
psubspset.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
psubspi (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)
Distinct variable groups:   𝐴,𝑟,𝑞   𝐾,𝑞,𝑟   𝑋,𝑞,𝑟   𝐴,𝑞   𝑃,𝑞,𝑟
Allowed substitution hints:   𝐷(𝑟,𝑞)   𝑆(𝑟,𝑞)   (𝑟,𝑞)   (𝑟,𝑞)

Proof of Theorem psubspi
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 psubspset.l . . . . . 6 = (le‘𝐾)
2 psubspset.j . . . . . 6 = (join‘𝐾)
3 psubspset.a . . . . . 6 𝐴 = (Atoms‘𝐾)
4 psubspset.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
51, 2, 3, 4ispsubsp2 34512 . . . . 5 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
65simplbda 653 . . . 4 ((𝐾𝐷𝑋𝑆) → ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
76ex 450 . . 3 (𝐾𝐷 → (𝑋𝑆 → ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋)))
8 breq1 4616 . . . . . 6 (𝑝 = 𝑃 → (𝑝 (𝑞 𝑟) ↔ 𝑃 (𝑞 𝑟)))
982rexbidv 3050 . . . . 5 (𝑝 = 𝑃 → (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) ↔ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)))
10 eleq1 2686 . . . . 5 (𝑝 = 𝑃 → (𝑝𝑋𝑃𝑋))
119, 10imbi12d 334 . . . 4 (𝑝 = 𝑃 → ((∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ (∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟) → 𝑃𝑋)))
1211rspccv 3292 . . 3 (∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) → (𝑃𝐴 → (∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟) → 𝑃𝑋)))
137, 12syl6 35 . 2 (𝐾𝐷 → (𝑋𝑆 → (𝑃𝐴 → (∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟) → 𝑃𝑋))))
14133imp1 1277 1 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908   ⊆ wss 3555   class class class wbr 4613  ‘cfv 5847  (class class class)co 6604  lecple 15869  joincjn 16865  Atomscatm 34030  PSubSpcpsubsp 34262 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-psubsp 34269 This theorem is referenced by:  psubspi2N  34514  paddidm  34607
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