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Theorem osumclN 39141
Description: Closure of orthogonal sum. If 𝑋 and π‘Œ are orthogonal closed projective subspaces, then their sum is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
osumcl.p + = (+π‘ƒβ€˜πΎ)
osumcl.o βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
osumcl.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
osumclN (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (𝑋 + π‘Œ) ∈ 𝐢)

Proof of Theorem osumclN
StepHypRef Expression
1 simpl1 1191 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ 𝐾 ∈ HL)
2 simpl2 1192 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ 𝑋 ∈ 𝐢)
3 eqid 2732 . . . . 5 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
4 osumcl.c . . . . 5 𝐢 = (PSubClβ€˜πΎ)
53, 4psubclssatN 39115 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
61, 2, 5syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ 𝑋 βŠ† (Atomsβ€˜πΎ))
7 simpl3 1193 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ π‘Œ ∈ 𝐢)
83, 4psubclssatN 39115 . . . 4 ((𝐾 ∈ HL ∧ π‘Œ ∈ 𝐢) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
91, 7, 8syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ π‘Œ βŠ† (Atomsβ€˜πΎ))
10 osumcl.p . . . 4 + = (+π‘ƒβ€˜πΎ)
113, 10paddssat 38988 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† (Atomsβ€˜πΎ) ∧ π‘Œ βŠ† (Atomsβ€˜πΎ)) β†’ (𝑋 + π‘Œ) βŠ† (Atomsβ€˜πΎ))
121, 6, 9, 11syl3anc 1371 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (𝑋 + π‘Œ) βŠ† (Atomsβ€˜πΎ))
13 simpll1 1212 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) ∧ 𝑋 = βˆ…) β†’ 𝐾 ∈ HL)
14 oveq1 7418 . . . . . 6 (𝑋 = βˆ… β†’ (𝑋 + π‘Œ) = (βˆ… + π‘Œ))
153, 10padd02 38986 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘Œ βŠ† (Atomsβ€˜πΎ)) β†’ (βˆ… + π‘Œ) = π‘Œ)
161, 9, 15syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (βˆ… + π‘Œ) = π‘Œ)
1714, 16sylan9eqr 2794 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) ∧ 𝑋 = βˆ…) β†’ (𝑋 + π‘Œ) = π‘Œ)
18 simpll3 1214 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) ∧ 𝑋 = βˆ…) β†’ π‘Œ ∈ 𝐢)
1917, 18eqeltrd 2833 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) ∧ 𝑋 = βˆ…) β†’ (𝑋 + π‘Œ) ∈ 𝐢)
20 osumcl.o . . . . 5 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
2120, 4psubcli2N 39113 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 + π‘Œ) ∈ 𝐢) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ))) = (𝑋 + π‘Œ))
2213, 19, 21syl2anc 584 . . 3 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) ∧ 𝑋 = βˆ…) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ))) = (𝑋 + π‘Œ))
2310, 20, 4osumcllem11N 39140 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ∧ 𝑋 β‰  βˆ…)) β†’ (𝑋 + π‘Œ) = ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ))))
2423anassrs 468 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) ∧ 𝑋 β‰  βˆ…) β†’ (𝑋 + π‘Œ) = ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ))))
2524eqcomd 2738 . . 3 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) ∧ 𝑋 β‰  βˆ…) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ))) = (𝑋 + π‘Œ))
2622, 25pm2.61dane 3029 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ))) = (𝑋 + π‘Œ))
273, 20, 4ispsubclN 39111 . . 3 (𝐾 ∈ HL β†’ ((𝑋 + π‘Œ) ∈ 𝐢 ↔ ((𝑋 + π‘Œ) βŠ† (Atomsβ€˜πΎ) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ))) = (𝑋 + π‘Œ))))
281, 27syl 17 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ ((𝑋 + π‘Œ) ∈ 𝐢 ↔ ((𝑋 + π‘Œ) βŠ† (Atomsβ€˜πΎ) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜(𝑋 + π‘Œ))) = (𝑋 + π‘Œ))))
2912, 26, 28mpbir2and 711 1 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢) ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ (𝑋 + π‘Œ) ∈ 𝐢)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   βŠ† wss 3948  βˆ…c0 4322  β€˜cfv 6543  (class class class)co 7411  Atomscatm 38436  HLchlt 38523  +𝑃cpadd 38969  βŠ₯𝑃cpolN 39076  PSubClcpscN 39108
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-pss 3967  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-iin 5000  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-p1 18383  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-pmap 38678  df-padd 38970  df-polarityN 39077  df-psubclN 39109
This theorem is referenced by:  pmapojoinN  39142  pexmidN  39143
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