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Theorem pclbtwnN 38363
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 2737 . . . . . 6 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
5 pclid.s . . . . . 6 𝑆 = (PSubSpβ€˜πΎ)
64, 5psubssat 38220 . . . . 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 38360 . . . 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 38362 . . . 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 3985 . 2 (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ (π‘ˆβ€˜π‘Œ) βŠ† 𝑋)
141, 13eqssd 3962 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 3911  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:  pclfinN  38366
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