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Theorem pclun2N 34203
 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 1054 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝐾 ∈ HL)
2 eqid 2610 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
3 pclun2.s . . . . 5 𝑆 = (PSubSp‘𝐾)
42, 3psubssat 34058 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
543adant3 1074 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
62, 3psubssat 34058 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
763adant2 1073 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
8 pclun2.p . . . 4 + = (+𝑃𝐾)
9 pclun2.c . . . 4 𝑈 = (PCl‘𝐾)
102, 8, 9pclunN 34202 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑈‘(𝑋𝑌)) = (𝑈‘(𝑋 + 𝑌)))
111, 5, 7, 10syl3anc 1318 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑈‘(𝑋𝑌)) = (𝑈‘(𝑋 + 𝑌)))
123, 8paddclN 34146 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
133, 9pclidN 34200 . . 3 ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ∈ 𝑆) → (𝑈‘(𝑋 + 𝑌)) = (𝑋 + 𝑌))
141, 12, 13syl2anc 691 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑈‘(𝑋 + 𝑌)) = (𝑋 + 𝑌))
1511, 14eqtrd 2644 1 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑈‘(𝑋𝑌)) = (𝑋 + 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Atomscatm 33568  HLchlt 33655  PSubSpcpsubsp 33800  +𝑃cpadd 34099  PClcpclN 34191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-psubsp 33807  df-padd 34100  df-pclN 34192 This theorem is referenced by: (None)
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