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Theorem paddclN 33944
Description: The projective sum of two subspaces is a subspace. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
paddidm.s 𝑆 = (PSubSp‘𝐾)
paddidm.p + = (+𝑃𝐾)
Assertion
Ref Expression
paddclN ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Proof of Theorem paddclN
Dummy variables 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1053 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝐾 ∈ HL)
2 eqid 2604 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
3 paddidm.s . . . . 5 𝑆 = (PSubSp‘𝐾)
42, 3psubssat 33856 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
543adant3 1073 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
62, 3psubssat 33856 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
763adant2 1072 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
8 paddidm.p . . . 4 + = (+𝑃𝐾)
92, 8paddssat 33916 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾))
101, 5, 7, 9syl3anc 1317 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾))
11 olc 397 . . . . 5 ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → ((𝑝 ∈ (𝑋 + 𝑌) ∨ 𝑝 ∈ (𝑋 + 𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))))
12 eqid 2604 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
13 eqid 2604 . . . . . . . 8 (join‘𝐾) = (join‘𝐾)
1412, 13, 2, 8elpadd 33901 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ (Atoms‘𝐾) ∧ (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) → (𝑝 ∈ ((𝑋 + 𝑌) + (𝑋 + 𝑌)) ↔ ((𝑝 ∈ (𝑋 + 𝑌) ∨ 𝑝 ∈ (𝑋 + 𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)))))
151, 10, 10, 14syl3anc 1317 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑝 ∈ ((𝑋 + 𝑌) + (𝑋 + 𝑌)) ↔ ((𝑝 ∈ (𝑋 + 𝑌) ∨ 𝑝 ∈ (𝑋 + 𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)))))
162, 8padd4N 33942 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = ((𝑋 + 𝑋) + (𝑌 + 𝑌)))
171, 5, 7, 5, 7, 16syl122anc 1326 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = ((𝑋 + 𝑋) + (𝑌 + 𝑌)))
183, 8paddidm 33943 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝑆) → (𝑋 + 𝑋) = 𝑋)
19183adant3 1073 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑋) = 𝑋)
203, 8paddidm 33943 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑌𝑆) → (𝑌 + 𝑌) = 𝑌)
21203adant2 1072 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑌 + 𝑌) = 𝑌)
2219, 21oveq12d 6540 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → ((𝑋 + 𝑋) + (𝑌 + 𝑌)) = (𝑋 + 𝑌))
2317, 22eqtrd 2638 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = (𝑋 + 𝑌))
2423eleq2d 2667 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑝 ∈ ((𝑋 + 𝑌) + (𝑋 + 𝑌)) ↔ 𝑝 ∈ (𝑋 + 𝑌)))
2515, 24bitr3d 268 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (((𝑝 ∈ (𝑋 + 𝑌) ∨ 𝑝 ∈ (𝑋 + 𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟))) ↔ 𝑝 ∈ (𝑋 + 𝑌)))
2611, 25syl5ib 232 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝 ∈ (𝑋 + 𝑌)))
2726expd 450 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑝 ∈ (Atoms‘𝐾) → (∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌))))
2827ralrimiv 2942 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → ∀𝑝 ∈ (Atoms‘𝐾)(∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌)))
2912, 13, 2, 3ispsubsp2 33848 . . 3 (𝐾 ∈ HL → ((𝑋 + 𝑌) ∈ 𝑆 ↔ ((𝑋 + 𝑌) ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ (Atoms‘𝐾)(∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌)))))
30293ad2ant1 1074 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → ((𝑋 + 𝑌) ∈ 𝑆 ↔ ((𝑋 + 𝑌) ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ (Atoms‘𝐾)(∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌)))))
3110, 28, 30mpbir2and 958 1 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1975  wral 2890  wrex 2891  wss 3534   class class class wbr 4572  cfv 5785  (class class class)co 6522  lecple 15716  joincjn 16708  Atomscatm 33366  HLchlt 33453  PSubSpcpsubsp 33598  +𝑃cpadd 33897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-1st 7031  df-2nd 7032  df-preset 16692  df-poset 16710  df-plt 16722  df-lub 16738  df-glb 16739  df-join 16740  df-meet 16741  df-p0 16803  df-lat 16810  df-clat 16872  df-oposet 33279  df-ol 33281  df-oml 33282  df-covers 33369  df-ats 33370  df-atl 33401  df-cvlat 33425  df-hlat 33454  df-psubsp 33605  df-padd 33898
This theorem is referenced by:  pmodl42N  33953  pclun2N  34001
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