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Theorem pmodN 38419
Description: The modular law for projective subspaces. (Contributed by NM, 26-Mar-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
pmodN ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 ∩ (π‘Œ + (𝑋 ∩ 𝑍))) = ((𝑋 ∩ π‘Œ) + (𝑋 ∩ 𝑍)))
Proof of Theorem pmodN
StepHypRef Expression
1 incom 4181 . 2 (𝑋 ∩ ((𝑋 ∩ 𝑍) + π‘Œ)) = (((𝑋 ∩ 𝑍) + π‘Œ) ∩ 𝑋)
2 hllat 37931 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
32adantr 481 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ 𝐾 ∈ Lat)
4 simpr2 1195 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ π‘Œ βŠ† 𝐴)
5 inss2 4209 . . . . 5 (𝑋 ∩ 𝑍) βŠ† 𝑍
6 simpr3 1196 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ 𝑍 βŠ† 𝐴)
75, 6sstrid 3973 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 ∩ 𝑍) βŠ† 𝐴)
8 pmod.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
9 pmod.p . . . . 5 + = (+π‘ƒβ€˜πΎ)
108, 9paddcom 38382 . . . 4 ((𝐾 ∈ Lat ∧ π‘Œ βŠ† 𝐴 ∧ (𝑋 ∩ 𝑍) βŠ† 𝐴) β†’ (π‘Œ + (𝑋 ∩ 𝑍)) = ((𝑋 ∩ 𝑍) + π‘Œ))
113, 4, 7, 10syl3anc 1371 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (π‘Œ + (𝑋 ∩ 𝑍)) = ((𝑋 ∩ 𝑍) + π‘Œ))
1211ineq2d 4192 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 ∩ (π‘Œ + (𝑋 ∩ 𝑍))) = (𝑋 ∩ ((𝑋 ∩ 𝑍) + π‘Œ)))
13 incom 4181 . . . 4 (𝑋 ∩ π‘Œ) = (π‘Œ ∩ 𝑋)
1413oveq2i 7388 . . 3 ((𝑋 ∩ 𝑍) + (𝑋 ∩ π‘Œ)) = ((𝑋 ∩ 𝑍) + (π‘Œ ∩ 𝑋))
15 inss2 4209 . . . . 5 (𝑋 ∩ π‘Œ) βŠ† π‘Œ
1615, 4sstrid 3973 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 ∩ π‘Œ) βŠ† 𝐴)
178, 9paddcom 38382 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 ∩ π‘Œ) βŠ† 𝐴 ∧ (𝑋 ∩ 𝑍) βŠ† 𝐴) β†’ ((𝑋 ∩ π‘Œ) + (𝑋 ∩ 𝑍)) = ((𝑋 ∩ 𝑍) + (𝑋 ∩ π‘Œ)))
183, 16, 7, 17syl3anc 1371 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ ((𝑋 ∩ π‘Œ) + (𝑋 ∩ 𝑍)) = ((𝑋 ∩ 𝑍) + (𝑋 ∩ π‘Œ)))
19 simpr1 1194 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ 𝑋 ∈ 𝑆)
207, 4, 193jca 1128 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ ((𝑋 ∩ 𝑍) βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 ∈ 𝑆))
21 inss1 4208 . . . . 5 (𝑋 ∩ 𝑍) βŠ† 𝑋
22 pmod.s . . . . . 6 𝑆 = (PSubSpβ€˜πΎ)
238, 22, 9pmod1i 38417 . . . . 5 ((𝐾 ∈ HL ∧ ((𝑋 ∩ 𝑍) βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 ∈ 𝑆)) β†’ ((𝑋 ∩ 𝑍) βŠ† 𝑋 β†’ (((𝑋 ∩ 𝑍) + π‘Œ) ∩ 𝑋) = ((𝑋 ∩ 𝑍) + (π‘Œ ∩ 𝑋))))
2421, 23mpi 20 . . . 4 ((𝐾 ∈ HL ∧ ((𝑋 ∩ 𝑍) βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 ∈ 𝑆)) β†’ (((𝑋 ∩ 𝑍) + π‘Œ) ∩ 𝑋) = ((𝑋 ∩ 𝑍) + (π‘Œ ∩ 𝑋)))
2520, 24syldan 591 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (((𝑋 ∩ 𝑍) + π‘Œ) ∩ 𝑋) = ((𝑋 ∩ 𝑍) + (π‘Œ ∩ 𝑋)))
2614, 18, 253eqtr4a 2797 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ ((𝑋 ∩ π‘Œ) + (𝑋 ∩ 𝑍)) = (((𝑋 ∩ 𝑍) + π‘Œ) ∩ 𝑋))
271, 12, 263eqtr4a 2797 1 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 ∩ (π‘Œ + (𝑋 ∩ 𝑍))) = ((𝑋 ∩ π‘Œ) + (𝑋 ∩ 𝑍)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3927   βŠ† wss 3928  β€˜cfv 6516  (class class class)co 7377  Latclat 18349  Atomscatm 37831  HLchlt 37918  PSubSpcpsubsp 38065  +𝑃cpadd 38364
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 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-1st 7941  df-2nd 7942  df-proset 18213  df-poset 18231  df-plt 18248  df-lub 18264  df-glb 18265  df-join 18266  df-meet 18267  df-p0 18343  df-lat 18350  df-covers 37834  df-ats 37835  df-atl 37866  df-cvlat 37890  df-hlat 37919  df-psubsp 38072  df-padd 38365
This theorem is referenced by: (None)
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