Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pclbtwnN Structured version   Visualization version   GIF version
Theorem pclbtwnN 39281
Description: A projective subspace sandwiched between a set of atoms and the set's projective subspace closure equals the closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclid.s 𝑆 = (PSubSpβ€˜πΎ)
pclid.c π‘ˆ = (PClβ€˜πΎ)
Assertion
Ref Expression
pclbtwnN (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ 𝑋 = (π‘ˆβ€˜π‘Œ))
Proof of Theorem pclbtwnN
StepHypRef Expression
1 simprr 770 . 2 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))
2 simpll 764 . . . 4 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ 𝐾 ∈ 𝑉)
3 simprl 768 . . . 4 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ π‘Œ βŠ† 𝑋)
4 eqid 2726 . . . . . 6 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
5 pclid.s . . . . . 6 𝑆 = (PSubSpβ€˜πΎ)
64, 5psubssat 39138 . . . . 5 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
76adantr 480 . . . 4 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
8 pclid.c . . . . 5 π‘ˆ = (PClβ€˜πΎ)
94, 8pclssN 39278 . . . 4 ((𝐾 ∈ 𝑉 ∧ π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (Atomsβ€˜πΎ)) β†’ (π‘ˆβ€˜π‘Œ) βŠ† (π‘ˆβ€˜π‘‹))
102, 3, 7, 9syl3anc 1368 . . 3 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ (π‘ˆβ€˜π‘Œ) βŠ† (π‘ˆβ€˜π‘‹))
115, 8pclidN 39280 . . . 4 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) β†’ (π‘ˆβ€˜π‘‹) = 𝑋)
1211adantr 480 . . 3 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ (π‘ˆβ€˜π‘‹) = 𝑋)
1310, 12sseqtrd 4017 . 2 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ (π‘ˆβ€˜π‘Œ) βŠ† 𝑋)
141, 13eqssd 3994 1 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ 𝑋 = (π‘ˆβ€˜π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  β€˜cfv 6537  Atomscatm 38646  PSubSpcpsubsp 38880  PClcpclN 39271
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 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-psubsp 38887  df-pclN 39272
This theorem is referenced by:  pclfinN  39284
  Copyright terms: Public domain W3C validator