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 36062
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 = (PSubSp‘𝐾)
pclun2.p + = (+𝑃𝐾)
pclun2.c 𝑈 = (PCl‘𝐾)
Assertion
Ref Expression
pclun2N ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑈‘(𝑋𝑌)) = (𝑋 + 𝑌))

Proof of Theorem pclun2N
StepHypRef Expression
1 simp1 1127 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝐾 ∈ HL)
2 eqid 2778 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
3 pclun2.s . . . . 5 𝑆 = (PSubSp‘𝐾)
42, 3psubssat 35917 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
543adant3 1123 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
62, 3psubssat 35917 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
763adant2 1122 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
8 pclun2.p . . . 4 + = (+𝑃𝐾)
9 pclun2.c . . . 4 𝑈 = (PCl‘𝐾)
102, 8, 9pclunN 36061 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑈‘(𝑋𝑌)) = (𝑈‘(𝑋 + 𝑌)))
111, 5, 7, 10syl3anc 1439 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑈‘(𝑋𝑌)) = (𝑈‘(𝑋 + 𝑌)))
123, 8paddclN 36005 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
133, 9pclidN 36059 . . 3 ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ∈ 𝑆) → (𝑈‘(𝑋 + 𝑌)) = (𝑋 + 𝑌))
141, 12, 13syl2anc 579 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑈‘(𝑋 + 𝑌)) = (𝑋 + 𝑌))
1511, 14eqtrd 2814 1 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑈‘(𝑋𝑌)) = (𝑋 + 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1601  wcel 2107  cun 3790  wss 3792  cfv 6137  (class class class)co 6924  Atomscatm 35426  HLchlt 35513  PSubSpcpsubsp 35659  +𝑃cpadd 35958  PClcpclN 36050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-proset 17325  df-poset 17343  df-plt 17355  df-lub 17371  df-glb 17372  df-join 17373  df-meet 17374  df-p0 17436  df-lat 17443  df-clat 17505  df-oposet 35339  df-ol 35341  df-oml 35342  df-covers 35429  df-ats 35430  df-atl 35461  df-cvlat 35485  df-hlat 35514  df-psubsp 35666  df-padd 35959  df-pclN 36051
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator