Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  paddunssN Structured version   Visualization version   GIF version
Theorem paddunssN 38983
Description: Projective subspace sum includes the set union of its arguments. (Contributed by NM, 12-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
padd0.a 𝐴 = (Atomsβ€˜πΎ)
padd0.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
paddunssN ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑋 βˆͺ π‘Œ) βŠ† (𝑋 + π‘Œ))
Proof of Theorem paddunssN
Dummy variables π‘ž 𝑝 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssun1 4172 . 2 (𝑋 βˆͺ π‘Œ) βŠ† ((𝑋 βˆͺ π‘Œ) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)})
2 eqid 2731 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
3 eqid 2731 . . 3 (joinβ€˜πΎ) = (joinβ€˜πΎ)
4 padd0.a . . 3 𝐴 = (Atomsβ€˜πΎ)
5 padd0.p . . 3 + = (+π‘ƒβ€˜πΎ)
62, 3, 4, 5paddval 38973 . 2 ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑋 + π‘Œ) = ((𝑋 βˆͺ π‘Œ) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)}))
71, 6sseqtrrid 4035 1 ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑋 βˆͺ π‘Œ) βŠ† (𝑋 + π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆƒwrex 3069  {crab 3431   βˆͺ cun 3946   βŠ† wss 3948   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  lecple 17209  joincjn 18269  Atomscatm 38437  +𝑃cpadd 38970
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-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-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-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-padd 38971
This theorem is referenced by:  pclunN  39073  paddunN  39102  pclfinclN  39125
  Copyright terms: Public domain W3C validator