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Theorem pclbtwnN 38768
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 772 . 2 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))
2 simpll 766 . . . 4 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ 𝐾 ∈ 𝑉)
3 simprl 770 . . . 4 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ π‘Œ βŠ† 𝑋)
4 eqid 2733 . . . . . 6 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
5 pclid.s . . . . . 6 𝑆 = (PSubSpβ€˜πΎ)
64, 5psubssat 38625 . . . . 5 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
76adantr 482 . . . 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 38765 . . . 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 1372 . . 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 38767 . . . 4 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) β†’ (π‘ˆβ€˜π‘‹) = 𝑋)
1211adantr 482 . . 3 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ (π‘ˆβ€˜π‘‹) = 𝑋)
1310, 12sseqtrd 4023 . 2 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ (π‘ˆβ€˜π‘Œ) βŠ† 𝑋)
141, 13eqssd 4000 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 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  β€˜cfv 6544  Atomscatm 38133  PSubSpcpsubsp 38367  PClcpclN 38758
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-psubsp 38374  df-pclN 38759
This theorem is referenced by:  pclfinN  38771
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