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Theorem paddclN 39016
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 = (PSubSpβ€˜πΎ)
paddidm.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
paddclN ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
Proof of Theorem paddclN
Dummy variables 𝑝 π‘ž π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1136 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ 𝐾 ∈ HL)
2 eqid 2732 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
3 paddidm.s . . . . 5 𝑆 = (PSubSpβ€˜πΎ)
42, 3psubssat 38928 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
543adant3 1132 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
62, 3psubssat 38928 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝑆) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
763adant2 1131 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
8 paddidm.p . . . 4 + = (+π‘ƒβ€˜πΎ)
92, 8paddssat 38988 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ) ∧ π‘Œ βŠ† (Atomsβ€˜πΎ)) β†’ (𝑋 + π‘Œ) βŠ† (Atomsβ€˜πΎ))
101, 5, 7, 9syl3anc 1371 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑋 + π‘Œ) βŠ† (Atomsβ€˜πΎ))
11 olc 866 . . . . 5 ((𝑝 ∈ (Atomsβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (𝑋 + π‘Œ)βˆƒπ‘Ÿ ∈ (𝑋 + π‘Œ)𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)) β†’ ((𝑝 ∈ (𝑋 + π‘Œ) ∨ 𝑝 ∈ (𝑋 + π‘Œ)) ∨ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (𝑋 + π‘Œ)βˆƒπ‘Ÿ ∈ (𝑋 + π‘Œ)𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))))
12 eqid 2732 . . . . . . . 8 (leβ€˜πΎ) = (leβ€˜πΎ)
13 eqid 2732 . . . . . . . 8 (joinβ€˜πΎ) = (joinβ€˜πΎ)
1412, 13, 2, 8elpadd 38973 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 + π‘Œ) βŠ† (Atomsβ€˜πΎ) ∧ (𝑋 + π‘Œ) βŠ† (Atomsβ€˜πΎ)) β†’ (𝑝 ∈ ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)) ↔ ((𝑝 ∈ (𝑋 + π‘Œ) ∨ 𝑝 ∈ (𝑋 + π‘Œ)) ∨ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (𝑋 + π‘Œ)βˆƒπ‘Ÿ ∈ (𝑋 + π‘Œ)𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
151, 10, 10, 14syl3anc 1371 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑝 ∈ ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)) ↔ ((𝑝 ∈ (𝑋 + π‘Œ) ∨ 𝑝 ∈ (𝑋 + π‘Œ)) ∨ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (𝑋 + π‘Œ)βˆƒπ‘Ÿ ∈ (𝑋 + π‘Œ)𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
162, 8padd4N 39014 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 βŠ† (Atomsβ€˜πΎ) ∧ π‘Œ βŠ† (Atomsβ€˜πΎ)) ∧ (𝑋 βŠ† (Atomsβ€˜πΎ) ∧ π‘Œ βŠ† (Atomsβ€˜πΎ))) β†’ ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)) = ((𝑋 + 𝑋) + (π‘Œ + π‘Œ)))
171, 5, 7, 5, 7, 16syl122anc 1379 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)) = ((𝑋 + 𝑋) + (π‘Œ + π‘Œ)))
183, 8paddidm 39015 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) β†’ (𝑋 + 𝑋) = 𝑋)
19183adant3 1132 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑋 + 𝑋) = 𝑋)
203, 8paddidm 39015 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝑆) β†’ (π‘Œ + π‘Œ) = π‘Œ)
21203adant2 1131 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (π‘Œ + π‘Œ) = π‘Œ)
2219, 21oveq12d 7429 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ ((𝑋 + 𝑋) + (π‘Œ + π‘Œ)) = (𝑋 + π‘Œ))
2317, 22eqtrd 2772 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)) = (𝑋 + π‘Œ))
2423eleq2d 2819 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑝 ∈ ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)) ↔ 𝑝 ∈ (𝑋 + π‘Œ)))
2515, 24bitr3d 280 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (((𝑝 ∈ (𝑋 + π‘Œ) ∨ 𝑝 ∈ (𝑋 + π‘Œ)) ∨ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (𝑋 + π‘Œ)βˆƒπ‘Ÿ ∈ (𝑋 + π‘Œ)𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))) ↔ 𝑝 ∈ (𝑋 + π‘Œ)))
2611, 25imbitrid 243 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ ((𝑝 ∈ (Atomsβ€˜πΎ) ∧ βˆƒπ‘ž ∈ (𝑋 + π‘Œ)βˆƒπ‘Ÿ ∈ (𝑋 + π‘Œ)𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)) β†’ 𝑝 ∈ (𝑋 + π‘Œ)))
2726expd 416 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑝 ∈ (Atomsβ€˜πΎ) β†’ (βˆƒπ‘ž ∈ (𝑋 + π‘Œ)βˆƒπ‘Ÿ ∈ (𝑋 + π‘Œ)𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) β†’ 𝑝 ∈ (𝑋 + π‘Œ))))
2827ralrimiv 3145 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ βˆ€π‘ ∈ (Atomsβ€˜πΎ)(βˆƒπ‘ž ∈ (𝑋 + π‘Œ)βˆƒπ‘Ÿ ∈ (𝑋 + π‘Œ)𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) β†’ 𝑝 ∈ (𝑋 + π‘Œ)))
2912, 13, 2, 3ispsubsp2 38920 . . 3 (𝐾 ∈ HL β†’ ((𝑋 + π‘Œ) ∈ 𝑆 ↔ ((𝑋 + π‘Œ) βŠ† (Atomsβ€˜πΎ) ∧ βˆ€π‘ ∈ (Atomsβ€˜πΎ)(βˆƒπ‘ž ∈ (𝑋 + π‘Œ)βˆƒπ‘Ÿ ∈ (𝑋 + π‘Œ)𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) β†’ 𝑝 ∈ (𝑋 + π‘Œ)))))
30293ad2ant1 1133 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ ((𝑋 + π‘Œ) ∈ 𝑆 ↔ ((𝑋 + π‘Œ) βŠ† (Atomsβ€˜πΎ) ∧ βˆ€π‘ ∈ (Atomsβ€˜πΎ)(βˆƒπ‘ž ∈ (𝑋 + π‘Œ)βˆƒπ‘Ÿ ∈ (𝑋 + π‘Œ)𝑝(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) β†’ 𝑝 ∈ (𝑋 + π‘Œ)))))
3110, 28, 30mpbir2and 711 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3948   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  lecple 17208  joincjn 18268  Atomscatm 38436  HLchlt 38523  PSubSpcpsubsp 38670  +𝑃cpadd 38969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-proset 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p0 18382  df-lat 18389  df-clat 18456  df-oposet 38349  df-ol 38351  df-oml 38352  df-covers 38439  df-ats 38440  df-atl 38471  df-cvlat 38495  df-hlat 38524  df-psubsp 38677  df-padd 38970
This theorem is referenced by:  pmodl42N  39025  pclun2N  39073
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