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Theorem elpcliN 38359
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 1137 . . . . . 6 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ 𝐾 ∈ 𝑉)
2 simp2 1138 . . . . . . 7 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ 𝑋 βŠ† π‘Œ)
3 eqid 2737 . . . . . . . . 9 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
4 elpcli.s . . . . . . . . 9 𝑆 = (PSubSpβ€˜πΎ)
53, 4psubssat 38220 . . . . . . . 8 ((𝐾 ∈ 𝑉 ∧ π‘Œ ∈ 𝑆) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
653adant2 1132 . . . . . . 7 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
72, 6sstrd 3955 . . . . . 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 38356 . . . . . 6 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧})
101, 7, 9syl2anc 585 . . . . 5 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧})
1110eleq2d 2824 . . . 4 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) ↔ 𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧}))
12 elintrabg 4923 . . . . 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 3971 . . . . . . . 8 (𝑧 = π‘Œ β†’ (𝑋 βŠ† 𝑧 ↔ 𝑋 βŠ† π‘Œ))
16 eleq2 2827 . . . . . . . 8 (𝑧 = π‘Œ β†’ (𝑄 ∈ 𝑧 ↔ 𝑄 ∈ π‘Œ))
1715, 16imbi12d 345 . . . . . . 7 (𝑧 = π‘Œ β†’ ((𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧) ↔ (𝑋 βŠ† π‘Œ β†’ 𝑄 ∈ π‘Œ)))
1817rspccv 3579 . . . . . 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 408 . . . 4 ((𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (βˆ€π‘§ ∈ 𝑆 (𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧) β†’ 𝑄 ∈ π‘Œ))
21203adant1 1131 . . 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 408 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3408   βŠ† wss 3911  βˆ© cint 4908  β€˜cfv 6497  Atomscatm 37728  PSubSpcpsubsp 37962  PClcpclN 38353
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-psubsp 37969  df-pclN 38354
This theorem is referenced by:  pclfinclN  38416
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