Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elpcliN Structured version   Visualization version   GIF version
Theorem elpcliN 39398
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 2728 . . . . . . . . 9 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
4 elpcli.s . . . . . . . . 9 𝑆 = (PSubSpβ€˜πΎ)
53, 4psubssat 39259 . . . . . . . 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 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 39395 . . . . . 6 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧})
101, 7, 9syl2anc 582 . . . . 5 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧})
1110eleq2d 2815 . . . 4 ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) ↔ 𝑄 ∈ ∩ {𝑧 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑧}))
12 elintrabg 4968 . . . . 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, 13biimtrdi 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 2818 . . . . . . . 8 (𝑧 = π‘Œ β†’ (𝑄 ∈ 𝑧 ↔ 𝑄 ∈ π‘Œ))
1715, 16imbi12d 343 . . . . . . 7 (𝑧 = π‘Œ β†’ ((𝑋 βŠ† 𝑧 β†’ 𝑄 ∈ 𝑧) ↔ (𝑋 βŠ† π‘Œ β†’ 𝑄 ∈ π‘Œ)))
1817rspccv 3608 . . . . . 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 405 . . . 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 405 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430   βŠ† wss 3949  βˆ© cint 4953  β€˜cfv 6553  Atomscatm 38767  PSubSpcpsubsp 39001  PClcpclN 39392
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-psubsp 39008  df-pclN 39393
This theorem is referenced by:  pclfinclN  39455
  Copyright terms: Public domain W3C validator