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Theorem elpcliN 39276
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 1133 . . . . . 6 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ 𝐾 ∈ 𝑉)
2 simp2 1134 . . . . . . 7 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ 𝑋 βŠ† π‘Œ)
3 eqid 2726 . . . . . . . . 9 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
4 elpcli.s . . . . . . . . 9 𝑆 = (PSubSpβ€˜πΎ)
53, 4psubssat 39137 . . . . . . . 8 ((𝐾 ∈ 𝑉 ∧ π‘Œ ∈ 𝑆) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
653adant2 1128 . . . . . . 7 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
72, 6sstrd 3987 . . . . . 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 39273 . . . . . 6 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧})
101, 7, 9syl2anc 583 . . . . 5 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧})
1110eleq2d 2813 . . . 4 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) ↔ 𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧}))
12 elintrabg 4958 . . . . 5 (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧} β†’ (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧} ↔ βˆ€π‘§ ∈ 𝑆 (𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧)))
1312ibi 267 . . . 4 (𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧} β†’ βˆ€π‘§ ∈ 𝑆 (𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧))
1411, 13syl6bi 253 . . 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 4003 . . . . . . . 8 (𝑧 = π‘Œ β†’ (𝑋 βŠ† 𝑧 ↔ 𝑋 βŠ† π‘Œ))
16 eleq2 2816 . . . . . . . 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 3603 . . . . . 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 406 . . . 4 ((𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (βˆ€π‘§ ∈ 𝑆 (𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧) β†’ 𝑄 ∈ π‘Œ))
21203adant1 1127 . . 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 406 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {crab 3426   βŠ† wss 3943  βˆ© cint 4943  β€˜cfv 6536  Atomscatm 38645  PSubSpcpsubsp 38879  PClcpclN 39270
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-rep 5278  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-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  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-int 4944  df-iun 4992  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-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-psubsp 38886  df-pclN 39271
This theorem is referenced by:  pclfinclN  39333
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