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Theorem psubspi 39130
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 39129 . . . . 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 499 . . . 4 ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) β†’ βˆ€π‘ ∈ 𝐴 (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ 𝑋 𝑝 ≀ (π‘ž ∨ π‘Ÿ) β†’ 𝑝 ∈ 𝑋))
76ex 412 . . 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 5144 . . . . . 6 (𝑝 = 𝑃 β†’ (𝑝 ≀ (π‘ž ∨ π‘Ÿ) ↔ 𝑃 ≀ (π‘ž ∨ π‘Ÿ)))
982rexbidv 3213 . . . . 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 2815 . . . . 5 (𝑝 = 𝑃 β†’ (𝑝 ∈ 𝑋 ↔ 𝑃 ∈ 𝑋))
119, 10imbi12d 344 . . . 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 3603 . . 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 1344 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064   βŠ† wss 3943   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  lecple 17210  joincjn 18273  Atomscatm 38645  PSubSpcpsubsp 38879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-psubsp 38886
This theorem is referenced by:  psubspi2N  39131  paddidm  39224
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