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Theorem paddunssN 35617
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 3927 . 2 (𝑋𝑌) ⊆ ((𝑋𝑌) ∪ {𝑝𝐴 ∣ ∃𝑞𝑋𝑟𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)})
2 eqid 2771 . . 3 (le‘𝐾) = (le‘𝐾)
3 eqid 2771 . . 3 (join‘𝐾) = (join‘𝐾)
4 padd0.a . . 3 𝐴 = (Atoms‘𝐾)
5 padd0.p . . 3 + = (+𝑃𝐾)
62, 3, 4, 5paddval 35607 . 2 ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) = ((𝑋𝑌) ∪ {𝑝𝐴 ∣ ∃𝑞𝑋𝑟𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)}))
71, 6syl5sseqr 3803 1 ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋𝑌) ⊆ (𝑋 + 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  wrex 3062  {crab 3065  cun 3721  wss 3723   class class class wbr 4787  cfv 6030  (class class class)co 6796  lecple 16156  joincjn 17152  Atomscatm 35072  +𝑃cpadd 35604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-padd 35605
This theorem is referenced by:  pclunN  35707  paddunN  35736  pclfinclN  35759
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