Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pclun2N Structured version   Visualization version   GIF version

Theorem pclun2N 38365
Description: The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclun2.s 𝑆 = (PSubSpβ€˜πΎ)
pclun2.p + = (+π‘ƒβ€˜πΎ)
pclun2.c π‘ˆ = (PClβ€˜πΎ)
Assertion
Ref Expression
pclun2N ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (π‘ˆβ€˜(𝑋 βˆͺ π‘Œ)) = (𝑋 + π‘Œ))

Proof of Theorem pclun2N
StepHypRef Expression
1 simp1 1137 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ 𝐾 ∈ HL)
2 eqid 2737 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
3 pclun2.s . . . . 5 𝑆 = (PSubSpβ€˜πΎ)
42, 3psubssat 38220 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
543adant3 1133 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
62, 3psubssat 38220 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝑆) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
763adant2 1132 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
8 pclun2.p . . . 4 + = (+π‘ƒβ€˜πΎ)
9 pclun2.c . . . 4 π‘ˆ = (PClβ€˜πΎ)
102, 8, 9pclunN 38364 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ) ∧ π‘Œ βŠ† (Atomsβ€˜πΎ)) β†’ (π‘ˆβ€˜(𝑋 βˆͺ π‘Œ)) = (π‘ˆβ€˜(𝑋 + π‘Œ)))
111, 5, 7, 10syl3anc 1372 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (π‘ˆβ€˜(𝑋 βˆͺ π‘Œ)) = (π‘ˆβ€˜(𝑋 + π‘Œ)))
123, 8paddclN 38308 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
133, 9pclidN 38362 . . 3 ((𝐾 ∈ HL ∧ (𝑋 + π‘Œ) ∈ 𝑆) β†’ (π‘ˆβ€˜(𝑋 + π‘Œ)) = (𝑋 + π‘Œ))
141, 12, 13syl2anc 585 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (π‘ˆβ€˜(𝑋 + π‘Œ)) = (𝑋 + π‘Œ))
1511, 14eqtrd 2777 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (π‘ˆβ€˜(𝑋 βˆͺ π‘Œ)) = (𝑋 + π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βˆͺ cun 3909   βŠ† wss 3911  β€˜cfv 6497  (class class class)co 7358  Atomscatm 37728  HLchlt 37815  PSubSpcpsubsp 37962  +𝑃cpadd 38261  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  ax-un 7673
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-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-lat 18322  df-clat 18389  df-oposet 37641  df-ol 37643  df-oml 37644  df-covers 37731  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816  df-psubsp 37969  df-padd 38262  df-pclN 38354
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator