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Theorem psubspi 38618
Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012.)
Hypotheses
Ref Expression
psubspset.l ≀ = (leβ€˜πΎ)
psubspset.j ∨ = (joinβ€˜πΎ)
psubspset.a 𝐴 = (Atomsβ€˜πΎ)
psubspset.s 𝑆 = (PSubSpβ€˜πΎ)
Assertion
Ref Expression
psubspi (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑃 ≀ (π‘ž ∨ π‘Ÿ)) β†’ 𝑃 ∈ 𝑋)
Distinct variable groups:   𝐴,π‘Ÿ,π‘ž   𝐾,π‘ž,π‘Ÿ   𝑋,π‘ž,π‘Ÿ   𝐴,π‘ž   𝑃,π‘ž,π‘Ÿ
Allowed substitution hints:   𝐷(π‘Ÿ,π‘ž)   𝑆(π‘Ÿ,π‘ž)   ∨ (π‘Ÿ,π‘ž)   ≀ (π‘Ÿ,π‘ž)

Proof of Theorem psubspi
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 psubspset.l . . . . . 6 ≀ = (leβ€˜πΎ)
2 psubspset.j . . . . . 6 ∨ = (joinβ€˜πΎ)
3 psubspset.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
4 psubspset.s . . . . . 6 𝑆 = (PSubSpβ€˜πΎ)
51, 2, 3, 4ispsubsp2 38617 . . . . 5 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑆 ↔ (𝑋 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝐴 (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝 ≀ (π‘ž ∨ π‘Ÿ) β†’ 𝑝 ∈ 𝑋))))
65simplbda 501 . . . 4 ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) β†’ βˆ€π‘ ∈ 𝐴 (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝 ≀ (π‘ž ∨ π‘Ÿ) β†’ 𝑝 ∈ 𝑋))
76ex 414 . . 3 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑆 β†’ βˆ€π‘ ∈ 𝐴 (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝 ≀ (π‘ž ∨ π‘Ÿ) β†’ 𝑝 ∈ 𝑋)))
8 breq1 5152 . . . . . 6 (𝑝 = 𝑃 β†’ (𝑝 ≀ (π‘ž ∨ π‘Ÿ) ↔ 𝑃 ≀ (π‘ž ∨ π‘Ÿ)))
982rexbidv 3220 . . . . 5 (𝑝 = 𝑃 β†’ (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝 ≀ (π‘ž ∨ π‘Ÿ) ↔ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑃 ≀ (π‘ž ∨ π‘Ÿ)))
10 eleq1 2822 . . . . 5 (𝑝 = 𝑃 β†’ (𝑝 ∈ 𝑋 ↔ 𝑃 ∈ 𝑋))
119, 10imbi12d 345 . . . 4 (𝑝 = 𝑃 β†’ ((βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝 ≀ (π‘ž ∨ π‘Ÿ) β†’ 𝑝 ∈ 𝑋) ↔ (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑃 ≀ (π‘ž ∨ π‘Ÿ) β†’ 𝑃 ∈ 𝑋)))
1211rspccv 3610 . . 3 (βˆ€π‘ ∈ 𝐴 (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝 ≀ (π‘ž ∨ π‘Ÿ) β†’ 𝑝 ∈ 𝑋) β†’ (𝑃 ∈ 𝐴 β†’ (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑃 ≀ (π‘ž ∨ π‘Ÿ) β†’ 𝑃 ∈ 𝑋)))
137, 12syl6 35 . 2 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑆 β†’ (𝑃 ∈ 𝐴 β†’ (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑃 ≀ (π‘ž ∨ π‘Ÿ) β†’ 𝑃 ∈ 𝑋))))
14133imp1 1348 1 (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑃 ≀ (π‘ž ∨ π‘Ÿ)) β†’ 𝑃 ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   βŠ† wss 3949   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  lecple 17204  joincjn 18264  Atomscatm 38133  PSubSpcpsubsp 38367
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
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 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-psubsp 38374
This theorem is referenced by:  psubspi2N  38619  paddidm  38712
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