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Theorem pmodN 39026
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 4202 . 2 (𝑋 ∩ ((𝑋 ∩ 𝑍) + π‘Œ)) = (((𝑋 ∩ 𝑍) + π‘Œ) ∩ 𝑋)
2 hllat 38538 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
32adantr 479 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ 𝐾 ∈ Lat)
4 simpr2 1193 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ π‘Œ βŠ† 𝐴)
5 inss2 4230 . . . . 5 (𝑋 ∩ 𝑍) βŠ† 𝑍
6 simpr3 1194 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ 𝑍 βŠ† 𝐴)
75, 6sstrid 3994 . . . 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 38989 . . . 4 ((𝐾 ∈ Lat ∧ π‘Œ βŠ† 𝐴 ∧ (𝑋 ∩ 𝑍) βŠ† 𝐴) β†’ (π‘Œ + (𝑋 ∩ 𝑍)) = ((𝑋 ∩ 𝑍) + π‘Œ))
113, 4, 7, 10syl3anc 1369 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (π‘Œ + (𝑋 ∩ 𝑍)) = ((𝑋 ∩ 𝑍) + π‘Œ))
1211ineq2d 4213 . 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 4202 . . . 4 (𝑋 ∩ π‘Œ) = (π‘Œ ∩ 𝑋)
1413oveq2i 7424 . . 3 ((𝑋 ∩ 𝑍) + (𝑋 ∩ π‘Œ)) = ((𝑋 ∩ 𝑍) + (π‘Œ ∩ 𝑋))
15 inss2 4230 . . . . 5 (𝑋 ∩ π‘Œ) βŠ† π‘Œ
1615, 4sstrid 3994 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 ∩ π‘Œ) βŠ† 𝐴)
178, 9paddcom 38989 . . . 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 1369 . . 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 1192 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ 𝑋 ∈ 𝑆)
207, 4, 193jca 1126 . . . 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 4229 . . . . 5 (𝑋 ∩ 𝑍) βŠ† 𝑋
22 pmod.s . . . . . 6 𝑆 = (PSubSpβ€˜πΎ)
238, 22, 9pmod1i 39024 . . . . 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 589 . . 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 2796 . 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 2796 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ∩ cin 3948   βŠ† wss 3949  β€˜cfv 6544  (class class class)co 7413  Latclat 18390  Atomscatm 38438  HLchlt 38525  PSubSpcpsubsp 38672  +𝑃cpadd 38971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7979  df-2nd 7980  df-proset 18254  df-poset 18272  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-lat 18391  df-covers 38441  df-ats 38442  df-atl 38473  df-cvlat 38497  df-hlat 38526  df-psubsp 38679  df-padd 38972
This theorem is referenced by: (None)
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