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Theorem sspadd2 39153
Description: A projective subspace sum is a superset of its second summand. (ssun2 4173 analog.) (Contributed by NM, 3-Jan-2012.)
Hypotheses
Ref Expression
padd0.a 𝐴 = (Atomsβ€˜πΎ)
padd0.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
sspadd2 ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ 𝑋 βŠ† (π‘Œ + 𝑋))
Proof of Theorem sspadd2
Dummy variables π‘ž 𝑝 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssun2 4173 . . 3 𝑋 βŠ† (π‘Œ βˆͺ 𝑋)
2 ssun1 4172 . . 3 (π‘Œ βˆͺ 𝑋) βŠ† ((π‘Œ βˆͺ 𝑋) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ π‘Œ βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)})
31, 2sstri 3991 . 2 𝑋 βŠ† ((π‘Œ βˆͺ 𝑋) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ π‘Œ βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)})
4 eqid 2731 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
5 eqid 2731 . . . 4 (joinβ€˜πΎ) = (joinβ€˜πΎ)
6 padd0.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
7 padd0.p . . . 4 + = (+π‘ƒβ€˜πΎ)
84, 5, 6, 7paddval 39135 . . 3 ((𝐾 ∈ 𝐡 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘Œ + 𝑋) = ((π‘Œ βˆͺ 𝑋) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ π‘Œ βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)}))
983com23 1125 . 2 ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (π‘Œ + 𝑋) = ((π‘Œ βˆͺ 𝑋) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ π‘Œ βˆƒπ‘Ÿ ∈ 𝑋 𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)}))
103, 9sseqtrrid 4035 1 ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ 𝑋 βŠ† (π‘Œ + 𝑋))
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 17211  joincjn 18274  Atomscatm 38599  +𝑃cpadd 39132
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 39133
This theorem is referenced by:  paddasslem11  39167  paddasslem12  39168  paddssw2  39181  pmodlem2  39184  pmodl42N  39188  osumcllem10N  39302  pexmidlem7N  39313  pl42lem3N  39318
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