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Theorem pclun2N 39234
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 1135 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝐾 ∈ HL)
2 eqid 2731 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
3 pclun2.s . . . . 5 𝑆 = (PSubSp‘𝐾)
42, 3psubssat 39089 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
543adant3 1131 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
62, 3psubssat 39089 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
763adant2 1130 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
8 pclun2.p . . . 4 + = (+𝑃𝐾)
9 pclun2.c . . . 4 𝑈 = (PCl‘𝐾)
102, 8, 9pclunN 39233 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑈‘(𝑋𝑌)) = (𝑈‘(𝑋 + 𝑌)))
111, 5, 7, 10syl3anc 1370 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑈‘(𝑋𝑌)) = (𝑈‘(𝑋 + 𝑌)))
123, 8paddclN 39177 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
133, 9pclidN 39231 . . 3 ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ∈ 𝑆) → (𝑈‘(𝑋 + 𝑌)) = (𝑋 + 𝑌))
141, 12, 13syl2anc 583 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑈‘(𝑋 + 𝑌)) = (𝑋 + 𝑌))
1511, 14eqtrd 2771 1 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑈‘(𝑋𝑌)) = (𝑋 + 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1540  wcel 2105  cun 3946  wss 3948  cfv 6543  (class class class)co 7412  Atomscatm 38597  HLchlt 38684  PSubSpcpsubsp 38831  +𝑃cpadd 39130  PClcpclN 39222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-lat 18395  df-clat 18462  df-oposet 38510  df-ol 38512  df-oml 38513  df-covers 38600  df-ats 38601  df-atl 38632  df-cvlat 38656  df-hlat 38685  df-psubsp 38838  df-padd 39131  df-pclN 39223
This theorem is referenced by: (None)
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