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Theorem paddclN 39844
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 1137 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝐾 ∈ HL)
2 eqid 2737 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
3 paddidm.s . . . . 5 𝑆 = (PSubSp‘𝐾)
42, 3psubssat 39756 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
543adant3 1133 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
62, 3psubssat 39756 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
763adant2 1132 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
8 paddidm.p . . . 4 + = (+𝑃𝐾)
92, 8paddssat 39816 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾))
101, 5, 7, 9syl3anc 1373 . 2 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾))
11 olc 869 . . . . 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 2737 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
13 eqid 2737 . . . . . . . 8 (join‘𝐾) = (join‘𝐾)
1412, 13, 2, 8elpadd 39801 . . . . . . 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 1373 . . . . . 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 39842 . . . . . . . . 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 1381 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = ((𝑋 + 𝑋) + (𝑌 + 𝑌)))
183, 8paddidm 39843 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝑆) → (𝑋 + 𝑋) = 𝑋)
19183adant3 1133 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑋) = 𝑋)
203, 8paddidm 39843 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑌𝑆) → (𝑌 + 𝑌) = 𝑌)
21203adant2 1132 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑌 + 𝑌) = 𝑌)
2219, 21oveq12d 7449 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → ((𝑋 + 𝑋) + (𝑌 + 𝑌)) = (𝑋 + 𝑌))
2317, 22eqtrd 2777 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → ((𝑋 + 𝑌) + (𝑋 + 𝑌)) = (𝑋 + 𝑌))
2423eleq2d 2827 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑝 ∈ ((𝑋 + 𝑌) + (𝑋 + 𝑌)) ↔ 𝑝 ∈ (𝑋 + 𝑌)))
2515, 24bitr3d 281 . . . . 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, 25imbitrid 244 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)) → 𝑝 ∈ (𝑋 + 𝑌)))
2726expd 415 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑝 ∈ (Atoms‘𝐾) → (∃𝑞 ∈ (𝑋 + 𝑌)∃𝑟 ∈ (𝑋 + 𝑌)𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟) → 𝑝 ∈ (𝑋 + 𝑌))))
2827ralrimiv 3145 . 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 39748 . . 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 1134 . 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 713 1 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951   class class class wbr 5143  cfv 6561  (class class class)co 7431  lecple 17304  joincjn 18357  Atomscatm 39264  HLchlt 39351  PSubSpcpsubsp 39498  +𝑃cpadd 39797
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-psubsp 39505  df-padd 39798
This theorem is referenced by:  pmodl42N  39853  pclun2N  39901
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