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 39282
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 1133 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ 𝐾 ∈ HL)
2 eqid 2726 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
3 pclun2.s . . . . 5 𝑆 = (PSubSpβ€˜πΎ)
42, 3psubssat 39137 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
543adant3 1129 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
62, 3psubssat 39137 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝑆) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
763adant2 1128 . . 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 39281 . . 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 1368 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (π‘ˆβ€˜(𝑋 βˆͺ π‘Œ)) = (π‘ˆβ€˜(𝑋 + π‘Œ)))
123, 8paddclN 39225 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
133, 9pclidN 39279 . . 3 ((𝐾 ∈ HL ∧ (𝑋 + π‘Œ) ∈ 𝑆) β†’ (π‘ˆβ€˜(𝑋 + π‘Œ)) = (𝑋 + π‘Œ))
141, 12, 13syl2anc 583 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (π‘ˆβ€˜(𝑋 + π‘Œ)) = (𝑋 + π‘Œ))
1511, 14eqtrd 2766 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 1084   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3941   βŠ† wss 3943  β€˜cfv 6536  (class class class)co 7404  Atomscatm 38645  HLchlt 38732  PSubSpcpsubsp 38879  +𝑃cpadd 39178  PClcpclN 39270
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  ax-un 7721
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-rmo 3370  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 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-proset 18257  df-poset 18275  df-plt 18292  df-lub 18308  df-glb 18309  df-join 18310  df-meet 18311  df-p0 18387  df-lat 18394  df-clat 18461  df-oposet 38558  df-ol 38560  df-oml 38561  df-covers 38648  df-ats 38649  df-atl 38680  df-cvlat 38704  df-hlat 38733  df-psubsp 38886  df-padd 39179  df-pclN 39271
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator