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Theorem cdlemkuu 40877
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 6906 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑆𝑑) = (𝑆𝐷))
2 cdlemk3.o2 . . . . . . . . 9 𝑄 = (𝑆𝐷)
31, 2eqtr4di 2792 . . . . . . . 8 (𝑑 = 𝐷 → (𝑆𝑑) = 𝑄)
43fveq1d 6908 . . . . . . 7 (𝑑 = 𝐷 → ((𝑆𝑑)‘𝑃) = (𝑄𝑃))
5 cnveq 5886 . . . . . . . . 9 (𝑑 = 𝐷𝑑 = 𝐷)
65coeq2d 5875 . . . . . . . 8 (𝑑 = 𝐷 → (𝑒𝑑) = (𝑒𝐷))
76fveq2d 6910 . . . . . . 7 (𝑑 = 𝐷 → (𝑅‘(𝑒𝑑)) = (𝑅‘(𝑒𝐷)))
84, 7oveq12d 7448 . . . . . 6 (𝑑 = 𝐷 → (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))) = ((𝑄𝑃) (𝑅‘(𝑒𝐷))))
98oveq2d 7446 . . . . 5 (𝑑 = 𝐷 → ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑)))) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷)))))
109eqeq2d 2745 . . . 4 (𝑑 = 𝐷 → ((𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑)))) ↔ (𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷))))))
1110riotabidv 7389 . . 3 (𝑑 = 𝐷 → (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))) = (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷))))))
12 fveq2 6906 . . . . . . 7 (𝑒 = 𝐺 → (𝑅𝑒) = (𝑅𝐺))
1312oveq2d 7446 . . . . . 6 (𝑒 = 𝐺 → (𝑃 (𝑅𝑒)) = (𝑃 (𝑅𝐺)))
14 coeq1 5870 . . . . . . . 8 (𝑒 = 𝐺 → (𝑒𝐷) = (𝐺𝐷))
1514fveq2d 6910 . . . . . . 7 (𝑒 = 𝐺 → (𝑅‘(𝑒𝐷)) = (𝑅‘(𝐺𝐷)))
1615oveq2d 7446 . . . . . 6 (𝑒 = 𝐺 → ((𝑄𝑃) (𝑅‘(𝑒𝐷))) = ((𝑄𝑃) (𝑅‘(𝐺𝐷))))
1713, 16oveq12d 7448 . . . . 5 (𝑒 = 𝐺 → ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷)))) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷)))))
1817eqeq2d 2745 . . . 4 (𝑒 = 𝐺 → ((𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷)))) ↔ (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))))
1918riotabidv 7389 . . 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 7391 . . 3 (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))) ∈ V
2211, 19, 20, 21ovmpo 7592 . 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 40826 . . 3 (𝐺𝑇 → (𝑍𝐺) = (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))))
3332adantl 481 . 2 ((𝐷𝑇𝐺𝑇) → (𝑍𝐺) = (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))))
3422, 33eqtr4d 2777 1 ((𝐷𝑇𝐺𝑇) → (𝐷𝑌𝐺) = (𝑍𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  cmpt 5230  ccnv 5687  ccom 5692  cfv 6562  crio 7386  (class class class)co 7430  cmpo 7432  Basecbs 17244  lecple 17304  joincjn 18368  meetcmee 18369  Atomscatm 39244  LHypclh 39966  LTrncltrn 40083  trLctrl 40140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435
This theorem is referenced by:  cdlemk31  40878  cdlemkuel-3  40880  cdlemkuv2-3N  40881  cdlemk18-3N  40882  cdlemk22-3  40883  cdlemkyu  40909
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