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Theorem elpcliN 38759
Description: Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpcli.s 𝑆 = (PSubSpβ€˜πΎ)
elpcli.c π‘ˆ = (PClβ€˜πΎ)
Assertion
Ref Expression
elpcliN (((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) ∧ 𝑄 ∈ (π‘ˆβ€˜π‘‹)) β†’ 𝑄 ∈ π‘Œ)
Proof of Theorem elpcliN
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simp1 1136 . . . . . 6 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ 𝐾 ∈ 𝑉)
2 simp2 1137 . . . . . . 7 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ 𝑋 βŠ† π‘Œ)
3 eqid 2732 . . . . . . . . 9 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
4 elpcli.s . . . . . . . . 9 𝑆 = (PSubSpβ€˜πΎ)
53, 4psubssat 38620 . . . . . . . 8 ((𝐾 ∈ 𝑉 ∧ π‘Œ ∈ 𝑆) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
653adant2 1131 . . . . . . 7 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
72, 6sstrd 3992 . . . . . 6 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
8 elpcli.c . . . . . . 7 π‘ˆ = (PClβ€˜πΎ)
93, 4, 8pclvalN 38756 . . . . . 6 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧})
101, 7, 9syl2anc 584 . . . . 5 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧})
1110eleq2d 2819 . . . 4 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) ↔ 𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧}))
12 elintrabg 4965 . . . . 5 (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧} β†’ (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧} ↔ βˆ€π‘§ ∈ 𝑆 (𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧)))
1312ibi 266 . . . 4 (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧} β†’ βˆ€π‘§ ∈ 𝑆 (𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧))
1411, 13syl6bi 252 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) β†’ βˆ€π‘§ ∈ 𝑆 (𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧)))
15 sseq2 4008 . . . . . . . 8 (𝑧 = π‘Œ β†’ (𝑋 βŠ† 𝑧 ↔ 𝑋 βŠ† π‘Œ))
16 eleq2 2822 . . . . . . . 8 (𝑧 = π‘Œ β†’ (𝑄 ∈ 𝑧 ↔ 𝑄 ∈ π‘Œ))
1715, 16imbi12d 344 . . . . . . 7 (𝑧 = π‘Œ β†’ ((𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧) ↔ (𝑋 βŠ† π‘Œ β†’ 𝑄 ∈ π‘Œ)))
1817rspccv 3609 . . . . . 6 (βˆ€π‘§ ∈ 𝑆 (𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧) β†’ (π‘Œ ∈ 𝑆 β†’ (𝑋 βŠ† π‘Œ β†’ 𝑄 ∈ π‘Œ)))
1918com13 88 . . . . 5 (𝑋 βŠ† π‘Œ β†’ (π‘Œ ∈ 𝑆 β†’ (βˆ€π‘§ ∈ 𝑆 (𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧) β†’ 𝑄 ∈ π‘Œ)))
2019imp 407 . . . 4 ((𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (βˆ€π‘§ ∈ 𝑆 (𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧) β†’ 𝑄 ∈ π‘Œ))
21203adant1 1130 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (βˆ€π‘§ ∈ 𝑆 (𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧) β†’ 𝑄 ∈ π‘Œ))
2214, 21syld 47 . 2 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) β†’ 𝑄 ∈ π‘Œ))
2322imp 407 1 (((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) ∧ 𝑄 ∈ (π‘ˆβ€˜π‘‹)) β†’ 𝑄 ∈ π‘Œ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   βŠ† wss 3948  βˆ© cint 4950  β€˜cfv 6543  Atomscatm 38128  PSubSpcpsubsp 38362  PClcpclN 38753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-psubsp 38369  df-pclN 38754
This theorem is referenced by:  pclfinclN  38816
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