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Theorem ispsubsp 38258
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 38257 . . 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 2820 . 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 6860 . . . . 5 𝐴 ∈ V
87ssex 5282 . . . 4 (𝑋 βŠ† 𝐴 β†’ 𝑋 ∈ V)
98adantr 482 . . 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 3973 . . . 4 (π‘₯ = 𝑋 β†’ (π‘₯ βŠ† 𝐴 ↔ 𝑋 βŠ† 𝐴))
11 eleq2 2823 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (π‘Ÿ ∈ π‘₯ ↔ π‘Ÿ ∈ 𝑋))
1211imbi2d 341 . . . . . . 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 3306 . . . . 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 3306 . . . 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 632 . . 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 3642 . 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 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  Vcvv 3447   βŠ† wss 3914   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361  lecple 17148  joincjn 18208  Atomscatm 37775  PSubSpcpsubsp 38009
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-psubsp 38016
This theorem is referenced by:  ispsubsp2  38259  0psubN  38262  snatpsubN  38263  linepsubN  38265  atpsubN  38266  psubssat  38267  pmapsub  38281  pclclN  38404  pclfinN  38413
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