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Theorem cdlemkuu 36500
 Description: Convert between function and operation forms of 𝑌. TODO: Use operation form everywhere. (Contributed by NM, 6-Jul-2013.)
Hypotheses
Ref Expression
cdlemk3.b 𝐵 = (Base‘𝐾)
cdlemk3.l = (le‘𝐾)
cdlemk3.j = (join‘𝐾)
cdlemk3.m = (meet‘𝐾)
cdlemk3.a 𝐴 = (Atoms‘𝐾)
cdlemk3.h 𝐻 = (LHyp‘𝐾)
cdlemk3.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk3.r 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk3.s 𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))
cdlemk3.u1 𝑌 = (𝑑𝑇, 𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))))
cdlemk3.o2 𝑄 = (𝑆𝐷)
cdlemk3.u2 𝑍 = (𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷))))))
Assertion
Ref Expression
cdlemkuu ((𝐷𝑇𝐺𝑇) → (𝐷𝑌𝐺) = (𝑍𝐺))
Distinct variable groups:   𝑒,𝑑,𝑓,𝑖,   ,𝑖   ,𝑑,𝑒,𝑓,𝑖   𝐴,𝑖   𝑗,𝑑,𝐷,𝑒,𝑓,𝑖   𝑓,𝐹,𝑖   𝐺,𝑑,𝑒,𝑗   𝑖,𝐻   𝑖,𝐾   𝑓,𝑁,𝑖   𝑃,𝑑,𝑒,𝑓,𝑖   𝑄,𝑑,𝑒   𝑅,𝑑,𝑒,𝑓,𝑖   𝑇,𝑑,𝑒,𝑓,𝑖   𝑊,𝑑,𝑒,𝑓,𝑖
Allowed substitution hints:   𝐴(𝑒,𝑓,𝑗,𝑑)   𝐵(𝑒,𝑓,𝑖,𝑗,𝑑)   𝑃(𝑗)   𝑄(𝑓,𝑖,𝑗)   𝑅(𝑗)   𝑆(𝑒,𝑓,𝑖,𝑗,𝑑)   𝑇(𝑗)   𝐹(𝑒,𝑗,𝑑)   𝐺(𝑓,𝑖)   𝐻(𝑒,𝑓,𝑗,𝑑)   (𝑗)   𝐾(𝑒,𝑓,𝑗,𝑑)   (𝑒,𝑓,𝑗,𝑑)   (𝑗)   𝑁(𝑒,𝑗,𝑑)   𝑊(𝑗)   𝑌(𝑒,𝑓,𝑖,𝑗,𝑑)   𝑍(𝑒,𝑓,𝑖,𝑗,𝑑)

Proof of Theorem cdlemkuu
StepHypRef Expression
1 fveq2 6229 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑆𝑑) = (𝑆𝐷))
2 cdlemk3.o2 . . . . . . . . 9 𝑄 = (𝑆𝐷)
31, 2syl6eqr 2703 . . . . . . . 8 (𝑑 = 𝐷 → (𝑆𝑑) = 𝑄)
43fveq1d 6231 . . . . . . 7 (𝑑 = 𝐷 → ((𝑆𝑑)‘𝑃) = (𝑄𝑃))
5 cnveq 5328 . . . . . . . . 9 (𝑑 = 𝐷𝑑 = 𝐷)
65coeq2d 5317 . . . . . . . 8 (𝑑 = 𝐷 → (𝑒𝑑) = (𝑒𝐷))
76fveq2d 6233 . . . . . . 7 (𝑑 = 𝐷 → (𝑅‘(𝑒𝑑)) = (𝑅‘(𝑒𝐷)))
84, 7oveq12d 6708 . . . . . 6 (𝑑 = 𝐷 → (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))) = ((𝑄𝑃) (𝑅‘(𝑒𝐷))))
98oveq2d 6706 . . . . 5 (𝑑 = 𝐷 → ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑)))) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷)))))
109eqeq2d 2661 . . . 4 (𝑑 = 𝐷 → ((𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑)))) ↔ (𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷))))))
1110riotabidv 6653 . . 3 (𝑑 = 𝐷 → (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))) = (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷))))))
12 fveq2 6229 . . . . . . 7 (𝑒 = 𝐺 → (𝑅𝑒) = (𝑅𝐺))
1312oveq2d 6706 . . . . . 6 (𝑒 = 𝐺 → (𝑃 (𝑅𝑒)) = (𝑃 (𝑅𝐺)))
14 coeq1 5312 . . . . . . . 8 (𝑒 = 𝐺 → (𝑒𝐷) = (𝐺𝐷))
1514fveq2d 6233 . . . . . . 7 (𝑒 = 𝐺 → (𝑅‘(𝑒𝐷)) = (𝑅‘(𝐺𝐷)))
1615oveq2d 6706 . . . . . 6 (𝑒 = 𝐺 → ((𝑄𝑃) (𝑅‘(𝑒𝐷))) = ((𝑄𝑃) (𝑅‘(𝐺𝐷))))
1713, 16oveq12d 6708 . . . . 5 (𝑒 = 𝐺 → ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷)))) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷)))))
1817eqeq2d 2661 . . . 4 (𝑒 = 𝐺 → ((𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷)))) ↔ (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))))
1918riotabidv 6653 . . 3 (𝑒 = 𝐺 → (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷))))) = (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))))
20 cdlemk3.u1 . . 3 𝑌 = (𝑑𝑇, 𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))))
21 riotaex 6655 . . 3 (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))) ∈ V
2211, 19, 20, 21ovmpt2 6838 . 2 ((𝐷𝑇𝐺𝑇) → (𝐷𝑌𝐺) = (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))))
23 cdlemk3.b . . . 4 𝐵 = (Base‘𝐾)
24 cdlemk3.l . . . 4 = (le‘𝐾)
25 cdlemk3.j . . . 4 = (join‘𝐾)
26 cdlemk3.a . . . 4 𝐴 = (Atoms‘𝐾)
27 cdlemk3.h . . . 4 𝐻 = (LHyp‘𝐾)
28 cdlemk3.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
29 cdlemk3.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
30 cdlemk3.m . . . 4 = (meet‘𝐾)
31 cdlemk3.u2 . . . 4 𝑍 = (𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷))))))
3223, 24, 25, 26, 27, 28, 29, 30, 31cdlemksv 36449 . . 3 (𝐺𝑇 → (𝑍𝐺) = (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))))
3332adantl 481 . 2 ((𝐷𝑇𝐺𝑇) → (𝑍𝐺) = (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))))
3422, 33eqtr4d 2688 1 ((𝐷𝑇𝐺𝑇) → (𝐷𝑌𝐺) = (𝑍𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ↦ cmpt 4762  ◡ccnv 5142   ∘ ccom 5147  ‘cfv 5926  ℩crio 6650  (class class class)co 6690   ↦ cmpt2 6692  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  Atomscatm 34868  LHypclh 35588  LTrncltrn 35705  trLctrl 35763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695 This theorem is referenced by:  cdlemk31  36501  cdlemkuel-3  36503  cdlemkuv2-3N  36504  cdlemk18-3N  36505  cdlemk22-3  36506  cdlemkyu  36532
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