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Theorem pclbtwnN 39410
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 771 . 2 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))
2 simpll 765 . . . 4 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ 𝐾 ∈ 𝑉)
3 simprl 769 . . . 4 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ π‘Œ βŠ† 𝑋)
4 eqid 2728 . . . . . 6 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
5 pclid.s . . . . . 6 𝑆 = (PSubSpβ€˜πΎ)
64, 5psubssat 39267 . . . . 5 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
76adantr 479 . . . 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 39407 . . . 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 39409 . . . 4 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) β†’ (π‘ˆβ€˜π‘‹) = 𝑋)
1211adantr 479 . . 3 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ (π‘ˆβ€˜π‘‹) = 𝑋)
1310, 12sseqtrd 4022 . 2 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ (π‘ˆβ€˜π‘Œ) βŠ† 𝑋)
141, 13eqssd 3999 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 394   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949  β€˜cfv 6553  Atomscatm 38775  PSubSpcpsubsp 39009  PClcpclN 39400
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 39016  df-pclN 39401
This theorem is referenced by:  pclfinN  39413
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