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 37062
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 = (Atoms‘𝐾)
padd0.p + = (+𝑃𝐾)
Assertion
Ref Expression
paddunssN ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋𝑌) ⊆ (𝑋 + 𝑌))

Proof of Theorem paddunssN
Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssun1 4123 . 2 (𝑋𝑌) ⊆ ((𝑋𝑌) ∪ {𝑝𝐴 ∣ ∃𝑞𝑋𝑟𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)})
2 eqid 2822 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2822 . . 3 (join‘𝐾) = (join‘𝐾)
4 padd0.a . . 3 𝐴 = (Atoms‘𝐾)
5 padd0.p . . 3 + = (+𝑃𝐾)
62, 3, 4, 5paddval 37052 . 2 ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) = ((𝑋𝑌) ∪ {𝑝𝐴 ∣ ∃𝑞𝑋𝑟𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)}))
71, 6sseqtrrid 3995 1 ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋𝑌) ⊆ (𝑋 + 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2114  wrex 3131  {crab 3134  cun 3906  wss 3908   class class class wbr 5042  cfv 6334  (class class class)co 7140  lecple 16563  joincjn 17545  Atomscatm 36517  +𝑃cpadd 37049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-padd 37050
This theorem is referenced by:  pclunN  37152  paddunN  37181  pclfinclN  37204
  Copyright terms: Public domain W3C validator