Theorem paddunssN 36948

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	⊢ + = (+𝑃‘𝐾)

Assertion

Ref	Expression
paddunssN	⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ (𝑋 + 𝑌))

Proof of Theorem paddunssN

Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssun1 4151	. 2 ⊢ (𝑋 ∪ 𝑌) ⊆ ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)})
2		eqid 2824	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2824	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
4		padd0.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
5		padd0.p	. . 3 ⊢ + = (+𝑃‘𝐾)
6	2, 3, 4, 5	paddval 36938	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) = ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)}))
7	1, 6	sseqtrrid 4023	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ (𝑋 + 𝑌))

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∃wrex 3142  {crab 3145   ∪ cun 3937   ⊆ wss 3939   class class class wbr 5069  ‘cfv 6358  (class class class)co 7159  lecple 16575  joincjn 17557  Atomscatm 36403  +_𝑃cpadd 36935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-padd 36936

This theorem is referenced by:  pclunN  37038  paddunN  37067  pclfinclN  37090