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Theorem isdlat 17508
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 6411 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isdlat.b . . . . 5 𝐵 = (Base‘𝐾)
31, 2syl6eqr 2851 . . . 4 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6411 . . . . . 6 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
5 isdlat.j . . . . . 6 = (join‘𝐾)
64, 5syl6eqr 2851 . . . . 5 (𝑘 = 𝐾 → (join‘𝑘) = )
7 fveq2 6411 . . . . . . 7 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
8 isdlat.m . . . . . . 7 = (meet‘𝐾)
97, 8syl6eqr 2851 . . . . . 6 (𝑘 = 𝐾 → (meet‘𝑘) = )
109sbceq1d 3638 . . . . 5 (𝑘 = 𝐾 → ([(meet‘𝑘) / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ [ / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
116, 10sbceqbid 3640 . . . 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 3640 . . 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))))
132fvexi 6425 . . . 4 𝐵 ∈ V
145fvexi 6425 . . . 4 ∈ V
158fvexi 6425 . . . 4 ∈ V
16 raleq 3321 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
1716raleqbi1dv 3329 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
1817raleqbi1dv 3329 . . . . . 6 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
19 simpr 478 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → 𝑚 = )
20 eqidd 2800 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → 𝑥 = 𝑥)
21 simpl 475 . . . . . . . . . . 11 ((𝑗 = 𝑚 = ) → 𝑗 = )
2221oveqd 6895 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → (𝑦𝑗𝑧) = (𝑦 𝑧))
2319, 20, 22oveq123d 6899 . . . . . . . . 9 ((𝑗 = 𝑚 = ) → (𝑥𝑚(𝑦𝑗𝑧)) = (𝑥 (𝑦 𝑧)))
2419oveqd 6895 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → (𝑥𝑚𝑦) = (𝑥 𝑦))
2519oveqd 6895 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → (𝑥𝑚𝑧) = (𝑥 𝑧))
2621, 24, 25oveq123d 6899 . . . . . . . . 9 ((𝑗 = 𝑚 = ) → ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) = ((𝑥 𝑦) (𝑥 𝑧)))
2723, 26eqeq12d 2814 . . . . . . . 8 ((𝑗 = 𝑚 = ) → ((𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
2827ralbidv 3167 . . . . . . 7 ((𝑗 = 𝑚 = ) → (∀𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
29282ralbidv 3170 . . . . . 6 ((𝑗 = 𝑚 = ) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
3018, 29sylan9bb 506 . . . . 5 ((𝑏 = 𝐵 ∧ (𝑗 = 𝑚 = )) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
31303impb 1144 . . . 4 ((𝑏 = 𝐵𝑗 = 𝑚 = ) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
3213, 14, 15, 31sbc3ie 3703 . . 3 ([𝐵 / 𝑏][ / 𝑗][ / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧)))
3312, 32syl6bb 279 . 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))))
34 df-dlat 17507 . 2 DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))}
3533, 34elrab2 3560 1 (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  [wsbc 3633  cfv 6101  (class class class)co 6878  Basecbs 16184  joincjn 17259  meetcmee 17260  Latclat 17360  DLatcdlat 17506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881  df-dlat 17507
This theorem is referenced by:  dlatmjdi  17509  dlatl  17510  odudlatb  17511
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