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Theorem isdlat 17797
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 6665 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isdlat.b . . . . 5 𝐵 = (Base‘𝐾)
31, 2syl6eqr 2874 . . . 4 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6665 . . . . . 6 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
5 isdlat.j . . . . . 6 = (join‘𝐾)
64, 5syl6eqr 2874 . . . . 5 (𝑘 = 𝐾 → (join‘𝑘) = )
7 fveq2 6665 . . . . . . 7 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
8 isdlat.m . . . . . . 7 = (meet‘𝐾)
97, 8syl6eqr 2874 . . . . . 6 (𝑘 = 𝐾 → (meet‘𝑘) = )
109sbceq1d 3777 . . . . 5 (𝑘 = 𝐾 → ([(meet‘𝑘) / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ [ / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
116, 10sbceqbid 3779 . . . 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 3779 . . 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 6679 . . . 4 𝐵 ∈ V
145fvexi 6679 . . . 4 ∈ V
158fvexi 6679 . . . 4 ∈ V
16 raleq 3406 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
1716raleqbi1dv 3404 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
1817raleqbi1dv 3404 . . . . . 6 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
19 simpr 487 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → 𝑚 = )
20 eqidd 2822 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → 𝑥 = 𝑥)
21 simpl 485 . . . . . . . . . . 11 ((𝑗 = 𝑚 = ) → 𝑗 = )
2221oveqd 7167 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → (𝑦𝑗𝑧) = (𝑦 𝑧))
2319, 20, 22oveq123d 7171 . . . . . . . . 9 ((𝑗 = 𝑚 = ) → (𝑥𝑚(𝑦𝑗𝑧)) = (𝑥 (𝑦 𝑧)))
2419oveqd 7167 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → (𝑥𝑚𝑦) = (𝑥 𝑦))
2519oveqd 7167 . . . . . . . . . 10 ((𝑗 = 𝑚 = ) → (𝑥𝑚𝑧) = (𝑥 𝑧))
2621, 24, 25oveq123d 7171 . . . . . . . . 9 ((𝑗 = 𝑚 = ) → ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) = ((𝑥 𝑦) (𝑥 𝑧)))
2723, 26eqeq12d 2837 . . . . . . . 8 ((𝑗 = 𝑚 = ) → ((𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
2827ralbidv 3197 . . . . . . 7 ((𝑗 = 𝑚 = ) → (∀𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
29282ralbidv 3199 . . . . . 6 ((𝑗 = 𝑚 = ) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
3018, 29sylan9bb 512 . . . . 5 ((𝑏 = 𝐵 ∧ (𝑗 = 𝑚 = )) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
31303impb 1111 . . . 4 ((𝑏 = 𝐵𝑗 = 𝑚 = ) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
3213, 14, 15, 31sbc3ie 3852 . . 3 ([𝐵 / 𝑏][ / 𝑗][ / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧)))
3312, 32syl6bb 289 . 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 17796 . 2 DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))}
3533, 34elrab2 3683 1 (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1533  wcel 2110  wral 3138  [wsbc 3772  cfv 6350  (class class class)co 7150  Basecbs 16477  joincjn 17548  meetcmee 17549  Latclat 17649  DLatcdlat 17795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-nul 5203
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-iota 6309  df-fv 6358  df-ov 7153  df-dlat 17796
This theorem is referenced by:  dlatmjdi  17798  dlatl  17799  odudlatb  17800
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