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Theorem psubspi 38556
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 38555 . . . . 5 (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))
65simplbda 501 . . . 4 ((𝐾𝐷𝑋𝑆) → ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))
76ex 414 . . 3 (𝐾𝐷 → (𝑋𝑆 → ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋)))
8 breq1 5150 . . . . . 6 (𝑝 = 𝑃 → (𝑝 (𝑞 𝑟) ↔ 𝑃 (𝑞 𝑟)))
982rexbidv 3220 . . . . 5 (𝑝 = 𝑃 → (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) ↔ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)))
10 eleq1 2822 . . . . 5 (𝑝 = 𝑃 → (𝑝𝑋𝑃𝑋))
119, 10imbi12d 345 . . . 4 (𝑝 = 𝑃 → ((∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋) ↔ (∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟) → 𝑃𝑋)))
1211rspccv 3609 . . 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 1348 1 (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  wss 3947   class class class wbr 5147  cfv 6540  (class class class)co 7404  lecple 17200  joincjn 18260  Atomscatm 38071  PSubSpcpsubsp 38305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407  df-psubsp 38312
This theorem is referenced by:  psubspi2N  38557  paddidm  38650
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