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Theorem pmod2iN 38806
Description: Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmod.a 𝐴 = (Atomsβ€˜πΎ)
pmod.s 𝑆 = (PSubSpβ€˜πΎ)
pmod.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
pmod2iN ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑍 βŠ† 𝑋 β†’ ((𝑋 ∩ π‘Œ) + 𝑍) = (𝑋 ∩ (π‘Œ + 𝑍))))
Proof of Theorem pmod2iN
StepHypRef Expression
1 incom 4201 . . . . . 6 (𝑋 ∩ π‘Œ) = (π‘Œ ∩ 𝑋)
21oveq1i 7421 . . . . 5 ((𝑋 ∩ π‘Œ) + 𝑍) = ((π‘Œ ∩ 𝑋) + 𝑍)
3 hllat 38319 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
433ad2ant1 1133 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ 𝐾 ∈ Lat)
5 simp22 1207 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ π‘Œ βŠ† 𝐴)
6 ssinss1 4237 . . . . . . 7 (π‘Œ βŠ† 𝐴 β†’ (π‘Œ ∩ 𝑋) βŠ† 𝐴)
75, 6syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ (π‘Œ ∩ 𝑋) βŠ† 𝐴)
8 simp23 1208 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ 𝑍 βŠ† 𝐴)
9 pmod.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
10 pmod.p . . . . . . 7 + = (+π‘ƒβ€˜πΎ)
119, 10paddcom 38770 . . . . . 6 ((𝐾 ∈ Lat ∧ (π‘Œ ∩ 𝑋) βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ ((π‘Œ ∩ 𝑋) + 𝑍) = (𝑍 + (π‘Œ ∩ 𝑋)))
124, 7, 8, 11syl3anc 1371 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ ((π‘Œ ∩ 𝑋) + 𝑍) = (𝑍 + (π‘Œ ∩ 𝑋)))
132, 12eqtrid 2784 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ ((𝑋 ∩ π‘Œ) + 𝑍) = (𝑍 + (π‘Œ ∩ 𝑋)))
14 simp21 1206 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ 𝑋 ∈ 𝑆)
158, 5, 143jca 1128 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ (𝑍 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 ∈ 𝑆))
16 pmod.s . . . . . . 7 𝑆 = (PSubSpβ€˜πΎ)
179, 16, 10pmod1i 38805 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑍 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 ∈ 𝑆)) β†’ (𝑍 βŠ† 𝑋 β†’ ((𝑍 + π‘Œ) ∩ 𝑋) = (𝑍 + (π‘Œ ∩ 𝑋))))
18173impia 1117 . . . . 5 ((𝐾 ∈ HL ∧ (𝑍 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ 𝑍 βŠ† 𝑋) β†’ ((𝑍 + π‘Œ) ∩ 𝑋) = (𝑍 + (π‘Œ ∩ 𝑋)))
1915, 18syld3an2 1411 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ ((𝑍 + π‘Œ) ∩ 𝑋) = (𝑍 + (π‘Œ ∩ 𝑋)))
209, 10paddcom 38770 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑍 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑍 + π‘Œ) = (π‘Œ + 𝑍))
214, 8, 5, 20syl3anc 1371 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ (𝑍 + π‘Œ) = (π‘Œ + 𝑍))
2221ineq1d 4211 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ ((𝑍 + π‘Œ) ∩ 𝑋) = ((π‘Œ + 𝑍) ∩ 𝑋))
2313, 19, 223eqtr2d 2778 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ ((𝑋 ∩ π‘Œ) + 𝑍) = ((π‘Œ + 𝑍) ∩ 𝑋))
24 incom 4201 . . 3 ((π‘Œ + 𝑍) ∩ 𝑋) = (𝑋 ∩ (π‘Œ + 𝑍))
2523, 24eqtrdi 2788 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ ((𝑋 ∩ π‘Œ) + 𝑍) = (𝑋 ∩ (π‘Œ + 𝑍)))
26253expia 1121 1 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑍 βŠ† 𝑋 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   ∩ cin 3947   βŠ† wss 3948  β€˜cfv 6543  (class class class)co 7411  Latclat 18386  Atomscatm 38219  HLchlt 38306  PSubSpcpsubsp 38453  +𝑃cpadd 38752
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 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-lat 18387  df-covers 38222  df-ats 38223  df-atl 38254  df-cvlat 38278  df-hlat 38307  df-psubsp 38460  df-padd 38753
This theorem is referenced by: (None)
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