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Theorem pmod2iN 38720
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 4202 . . . . . 6 (𝑋 ∩ π‘Œ) = (π‘Œ ∩ 𝑋)
21oveq1i 7419 . . . . 5 ((𝑋 ∩ π‘Œ) + 𝑍) = ((π‘Œ ∩ 𝑋) + 𝑍)
3 hllat 38233 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
433ad2ant1 1134 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ 𝐾 ∈ Lat)
5 simp22 1208 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ π‘Œ βŠ† 𝐴)
6 ssinss1 4238 . . . . . . 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 1209 . . . . . 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 38684 . . . . . 6 ((𝐾 ∈ Lat ∧ (π‘Œ ∩ 𝑋) βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ ((π‘Œ ∩ 𝑋) + 𝑍) = (𝑍 + (π‘Œ ∩ 𝑋)))
124, 7, 8, 11syl3anc 1372 . . . . 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 2785 . . . 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 1207 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ 𝑋 ∈ 𝑆)
158, 5, 143jca 1129 . . . . 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 38719 . . . . . 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 1118 . . . . 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 1412 . . . 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 38684 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑍 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑍 + π‘Œ) = (π‘Œ + 𝑍))
214, 8, 5, 20syl3anc 1372 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ 𝑍 βŠ† 𝑋) β†’ (𝑍 + π‘Œ) = (π‘Œ + 𝑍))
2221ineq1d 4212 . . . 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 2779 . . 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 4202 . . 3 ((π‘Œ + 𝑍) ∩ 𝑋) = (𝑋 ∩ (π‘Œ + 𝑍))
2523, 24eqtrdi 2789 . 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 1122 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∩ cin 3948   βŠ† wss 3949  β€˜cfv 6544  (class class class)co 7409  Latclat 18384  Atomscatm 38133  HLchlt 38220  PSubSpcpsubsp 38367  +𝑃cpadd 38666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  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 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-lat 18385  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-psubsp 38374  df-padd 38667
This theorem is referenced by: (None)
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