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Theorem ispsubsp 39127
Description: The predicate "is a projective subspace". (Contributed by NM, 2-Oct-2011.)
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
ispsubsp (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑆 ↔ (𝑋 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑋 βˆ€π‘ž ∈ 𝑋 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑋))))
Distinct variable groups:   𝐴,π‘Ÿ   π‘ž,𝑝,π‘Ÿ,𝐾   𝑋,𝑝,π‘ž,π‘Ÿ
Allowed substitution hints:   𝐴(π‘ž,𝑝)   𝐷(π‘Ÿ,π‘ž,𝑝)   𝑆(π‘Ÿ,π‘ž,𝑝)   ∨ (π‘Ÿ,π‘ž,𝑝)   ≀ (π‘Ÿ,π‘ž,𝑝)

Proof of Theorem ispsubsp
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 psubspset.l . . . 4 ≀ = (leβ€˜πΎ)
2 psubspset.j . . . 4 ∨ = (joinβ€˜πΎ)
3 psubspset.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
4 psubspset.s . . . 4 𝑆 = (PSubSpβ€˜πΎ)
51, 2, 3, 4psubspset 39126 . . 3 (𝐾 ∈ 𝐷 β†’ 𝑆 = {π‘₯ ∣ (π‘₯ βŠ† 𝐴 ∧ βˆ€π‘ ∈ π‘₯ βˆ€π‘ž ∈ π‘₯ βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ π‘₯))})
65eleq2d 2813 . 2 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ {π‘₯ ∣ (π‘₯ βŠ† 𝐴 ∧ βˆ€π‘ ∈ π‘₯ βˆ€π‘ž ∈ π‘₯ βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ π‘₯))}))
73fvexi 6898 . . . . 5 𝐴 ∈ V
87ssex 5314 . . . 4 (𝑋 βŠ† 𝐴 β†’ 𝑋 ∈ V)
98adantr 480 . . 3 ((𝑋 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑋 βˆ€π‘ž ∈ 𝑋 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑋)) β†’ 𝑋 ∈ V)
10 sseq1 4002 . . . 4 (π‘₯ = 𝑋 β†’ (π‘₯ βŠ† 𝐴 ↔ 𝑋 βŠ† 𝐴))
11 eleq2 2816 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (π‘Ÿ ∈ π‘₯ ↔ π‘Ÿ ∈ 𝑋))
1211imbi2d 340 . . . . . . 7 (π‘₯ = 𝑋 β†’ ((π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ π‘₯) ↔ (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑋)))
1312ralbidv 3171 . . . . . 6 (π‘₯ = 𝑋 β†’ (βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ π‘₯) ↔ βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑋)))
1413raleqbi1dv 3327 . . . . 5 (π‘₯ = 𝑋 β†’ (βˆ€π‘ž ∈ π‘₯ βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ π‘₯) ↔ βˆ€π‘ž ∈ 𝑋 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑋)))
1514raleqbi1dv 3327 . . . 4 (π‘₯ = 𝑋 β†’ (βˆ€π‘ ∈ π‘₯ βˆ€π‘ž ∈ π‘₯ βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ π‘₯) ↔ βˆ€π‘ ∈ 𝑋 βˆ€π‘ž ∈ 𝑋 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑋)))
1610, 15anbi12d 630 . . 3 (π‘₯ = 𝑋 β†’ ((π‘₯ βŠ† 𝐴 ∧ βˆ€π‘ ∈ π‘₯ βˆ€π‘ž ∈ π‘₯ βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ π‘₯)) ↔ (𝑋 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑋 βˆ€π‘ž ∈ 𝑋 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑋))))
179, 16elab3 3671 . 2 (𝑋 ∈ {π‘₯ ∣ (π‘₯ βŠ† 𝐴 ∧ βˆ€π‘ ∈ π‘₯ βˆ€π‘ž ∈ π‘₯ βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ π‘₯))} ↔ (𝑋 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑋 βˆ€π‘ž ∈ 𝑋 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑋)))
186, 17bitrdi 287 1 (𝐾 ∈ 𝐷 β†’ (𝑋 ∈ 𝑆 ↔ (𝑋 βŠ† 𝐴 ∧ βˆ€π‘ ∈ 𝑋 βˆ€π‘ž ∈ 𝑋 βˆ€π‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ (𝑝 ∨ π‘ž) β†’ π‘Ÿ ∈ 𝑋))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆ€wral 3055  Vcvv 3468   βŠ† wss 3943   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  lecple 17211  joincjn 18274  Atomscatm 38644  PSubSpcpsubsp 38878
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 38885
This theorem is referenced by:  ispsubsp2  39128  0psubN  39131  snatpsubN  39132  linepsubN  39134  atpsubN  39135  psubssat  39136  pmapsub  39150  pclclN  39273  pclfinN  39282
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