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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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