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Theorem isdlat 17114
Description: Property of being a distributive lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypotheses
Ref Expression
isdlat.b 𝐵 = (Base‘𝐾)
isdlat.j = (join‘𝐾)
isdlat.m = (meet‘𝐾)
Assertion
Ref Expression
isdlat (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐾   𝑥,𝐵,𝑦,𝑧   𝑥, ,𝑦,𝑧   𝑥, ,𝑦,𝑧

Proof of Theorem isdlat
Dummy variables 𝑘 𝑏 𝑗 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6148 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isdlat.b . . . . 5 𝐵 = (Base‘𝐾)
31, 2syl6eqr 2673 . . . 4 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6148 . . . . . 6 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
5 isdlat.j . . . . . 6 = (join‘𝐾)
64, 5syl6eqr 2673 . . . . 5 (𝑘 = 𝐾 → (join‘𝑘) = )
7 fveq2 6148 . . . . . . 7 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
8 isdlat.m . . . . . . 7 = (meet‘𝐾)
97, 8syl6eqr 2673 . . . . . 6 (𝑘 = 𝐾 → (meet‘𝑘) = )
109sbceq1d 3422 . . . . 5 (𝑘 = 𝐾 → ([(meet‘𝑘) / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ [ / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
116, 10sbceqbid 3424 . . . 4 (𝑘 = 𝐾 → ([(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ [ / 𝑗][ / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
123, 11sbceqbid 3424 . . 3 (𝑘 = 𝐾 → ([(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ [𝐵 / 𝑏][ / 𝑗][ / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
13 fvex 6158 . . . . 5 (Base‘𝐾) ∈ V
142, 13eqeltri 2694 . . . 4 𝐵 ∈ V
15 fvex 6158 . . . . 5 (join‘𝐾) ∈ V
165, 15eqeltri 2694 . . . 4 ∈ V
17 fvex 6158 . . . . 5 (meet‘𝐾) ∈ V
188, 17eqeltri 2694 . . . 4 ∈ V
19 raleq 3127 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
2019raleqbi1dv 3135 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
2120raleqbi1dv 3135 . . . . . 6 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
22 simpr 477 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → 𝑚 = )
23 eqidd 2622 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → 𝑥 = 𝑥)
24 simpl 473 . . . . . . . . . . 11 ((𝑗 = 𝑚 = ) → 𝑗 = )
2524oveqd 6621 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → (𝑦𝑗𝑧) = (𝑦 𝑧))
2622, 23, 25oveq123d 6625 . . . . . . . . 9 ((𝑗 = 𝑚 = ) → (𝑥𝑚(𝑦𝑗𝑧)) = (𝑥 (𝑦 𝑧)))
2722oveqd 6621 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → (𝑥𝑚𝑦) = (𝑥 𝑦))
2822oveqd 6621 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → (𝑥𝑚𝑧) = (𝑥 𝑧))
2924, 27, 28oveq123d 6625 . . . . . . . . 9 ((𝑗 = 𝑚 = ) → ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) = ((𝑥 𝑦) (𝑥 𝑧)))
3026, 29eqeq12d 2636 . . . . . . . 8 ((𝑗 = 𝑚 = ) → ((𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
3130ralbidv 2980 . . . . . . 7 ((𝑗 = 𝑚 = ) → (∀𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
32312ralbidv 2983 . . . . . 6 ((𝑗 = 𝑚 = ) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
3321, 32sylan9bb 735 . . . . 5 ((𝑏 = 𝐵 ∧ (𝑗 = 𝑚 = )) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
34333impb 1257 . . . 4 ((𝑏 = 𝐵𝑗 = 𝑚 = ) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
3514, 16, 18, 34sbc3ie 3489 . . 3 ([𝐵 / 𝑏][ / 𝑗][ / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧)))
3612, 35syl6bb 276 . 2 (𝑘 = 𝐾 → ([(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
37 df-dlat 17113 . 2 DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))}
3836, 37elrab2 3348 1 (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  [wsbc 3417  cfv 5847  (class class class)co 6604  Basecbs 15781  joincjn 16865  meetcmee 16866  Latclat 16966  DLatcdlat 17112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-dlat 17113
This theorem is referenced by:  dlatmjdi  17115  dlatl  17116  odudlatb  17117
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