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Theorem paddass 39012
Description: Projective subspace sum is associative. Equation 16.2.1 of [MaedaMaeda] p. 68. In our version, the subspaces do not have to be nonempty. (Contributed by NM, 29-Dec-2011.)
Hypotheses
Ref Expression
paddass.a 𝐴 = (Atomsβ€˜πΎ)
paddass.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
paddass ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ ((𝑋 + π‘Œ) + 𝑍) = (𝑋 + (π‘Œ + 𝑍)))
Proof of Theorem paddass
StepHypRef Expression
1 simpl 483 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ 𝐾 ∈ HL)
2 simpr3 1196 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ 𝑍 βŠ† 𝐴)
3 simpr2 1195 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ π‘Œ βŠ† 𝐴)
4 simpr1 1194 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ 𝑋 βŠ† 𝐴)
5 paddass.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
6 paddass.p . . . . 5 + = (+π‘ƒβ€˜πΎ)
75, 6paddasslem18 39011 . . . 4 ((𝐾 ∈ HL ∧ (𝑍 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† 𝐴)) β†’ (𝑍 + (π‘Œ + 𝑋)) βŠ† ((𝑍 + π‘Œ) + 𝑋))
81, 2, 3, 4, 7syl13anc 1372 . . 3 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑍 + (π‘Œ + 𝑋)) βŠ† ((𝑍 + π‘Œ) + 𝑋))
9 hllat 38536 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
105, 6paddcom 38987 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑋 + π‘Œ) = (π‘Œ + 𝑋))
119, 10syl3an1 1163 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑋 + π‘Œ) = (π‘Œ + 𝑋))
12113adant3r3 1184 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 + π‘Œ) = (π‘Œ + 𝑋))
1312oveq1d 7426 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ ((𝑋 + π‘Œ) + 𝑍) = ((π‘Œ + 𝑋) + 𝑍))
145, 6paddssat 38988 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘Œ + 𝑋) βŠ† 𝐴)
151, 3, 4, 14syl3anc 1371 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (π‘Œ + 𝑋) βŠ† 𝐴)
165, 6paddcom 38987 . . . . . 6 ((𝐾 ∈ Lat ∧ (π‘Œ + 𝑋) βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ ((π‘Œ + 𝑋) + 𝑍) = (𝑍 + (π‘Œ + 𝑋)))
179, 16syl3an1 1163 . . . . 5 ((𝐾 ∈ HL ∧ (π‘Œ + 𝑋) βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ ((π‘Œ + 𝑋) + 𝑍) = (𝑍 + (π‘Œ + 𝑋)))
181, 15, 2, 17syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ ((π‘Œ + 𝑋) + 𝑍) = (𝑍 + (π‘Œ + 𝑋)))
1913, 18eqtrd 2772 . . 3 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ ((𝑋 + π‘Œ) + 𝑍) = (𝑍 + (π‘Œ + 𝑋)))
205, 6paddcom 38987 . . . . . . 7 ((𝐾 ∈ Lat ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ (π‘Œ + 𝑍) = (𝑍 + π‘Œ))
219, 20syl3an1 1163 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ (π‘Œ + 𝑍) = (𝑍 + π‘Œ))
22213adant3r1 1182 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (π‘Œ + 𝑍) = (𝑍 + π‘Œ))
2322oveq2d 7427 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 + (π‘Œ + 𝑍)) = (𝑋 + (𝑍 + π‘Œ)))
245, 6paddssat 38988 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑍 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑍 + π‘Œ) βŠ† 𝐴)
251, 2, 3, 24syl3anc 1371 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑍 + π‘Œ) βŠ† 𝐴)
265, 6paddcom 38987 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋 βŠ† 𝐴 ∧ (𝑍 + π‘Œ) βŠ† 𝐴) β†’ (𝑋 + (𝑍 + π‘Œ)) = ((𝑍 + π‘Œ) + 𝑋))
279, 26syl3an1 1163 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ (𝑍 + π‘Œ) βŠ† 𝐴) β†’ (𝑋 + (𝑍 + π‘Œ)) = ((𝑍 + π‘Œ) + 𝑋))
281, 4, 25, 27syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 + (𝑍 + π‘Œ)) = ((𝑍 + π‘Œ) + 𝑋))
2923, 28eqtrd 2772 . . 3 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 + (π‘Œ + 𝑍)) = ((𝑍 + π‘Œ) + 𝑋))
308, 19, 293sstr4d 4029 . 2 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ ((𝑋 + π‘Œ) + 𝑍) βŠ† (𝑋 + (π‘Œ + 𝑍)))
315, 6paddasslem18 39011 . 2 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 + (π‘Œ + 𝑍)) βŠ† ((𝑋 + π‘Œ) + 𝑍))
3230, 31eqssd 3999 1 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ ((𝑋 + π‘Œ) + 𝑍) = (𝑋 + (π‘Œ + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  β€˜cfv 6543  (class class class)co 7411  Latclat 18388  Atomscatm 38436  HLchlt 38523  +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-padd 38970
This theorem is referenced by:  padd12N  39013  padd4N  39014  pmodl42N  39025  pmapjlln1  39029
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