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Theorem cdlemkuu 38149
 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 6652 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑆𝑑) = (𝑆𝐷))
2 cdlemk3.o2 . . . . . . . . 9 𝑄 = (𝑆𝐷)
31, 2eqtr4di 2875 . . . . . . . 8 (𝑑 = 𝐷 → (𝑆𝑑) = 𝑄)
43fveq1d 6654 . . . . . . 7 (𝑑 = 𝐷 → ((𝑆𝑑)‘𝑃) = (𝑄𝑃))
5 cnveq 5721 . . . . . . . . 9 (𝑑 = 𝐷𝑑 = 𝐷)
65coeq2d 5710 . . . . . . . 8 (𝑑 = 𝐷 → (𝑒𝑑) = (𝑒𝐷))
76fveq2d 6656 . . . . . . 7 (𝑑 = 𝐷 → (𝑅‘(𝑒𝑑)) = (𝑅‘(𝑒𝐷)))
84, 7oveq12d 7158 . . . . . 6 (𝑑 = 𝐷 → (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))) = ((𝑄𝑃) (𝑅‘(𝑒𝐷))))
98oveq2d 7156 . . . . 5 (𝑑 = 𝐷 → ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑)))) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷)))))
109eqeq2d 2833 . . . 4 (𝑑 = 𝐷 → ((𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑)))) ↔ (𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷))))))
1110riotabidv 7100 . . 3 (𝑑 = 𝐷 → (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))) = (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷))))))
12 fveq2 6652 . . . . . . 7 (𝑒 = 𝐺 → (𝑅𝑒) = (𝑅𝐺))
1312oveq2d 7156 . . . . . 6 (𝑒 = 𝐺 → (𝑃 (𝑅𝑒)) = (𝑃 (𝑅𝐺)))
14 coeq1 5705 . . . . . . . 8 (𝑒 = 𝐺 → (𝑒𝐷) = (𝐺𝐷))
1514fveq2d 6656 . . . . . . 7 (𝑒 = 𝐺 → (𝑅‘(𝑒𝐷)) = (𝑅‘(𝐺𝐷)))
1615oveq2d 7156 . . . . . 6 (𝑒 = 𝐺 → ((𝑄𝑃) (𝑅‘(𝑒𝐷))) = ((𝑄𝑃) (𝑅‘(𝐺𝐷))))
1713, 16oveq12d 7158 . . . . 5 (𝑒 = 𝐺 → ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷)))) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷)))))
1817eqeq2d 2833 . . . 4 (𝑒 = 𝐺 → ((𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝐷)))) ↔ (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))))
1918riotabidv 7100 . . 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 7102 . . 3 (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))) ∈ V
2211, 19, 20, 21ovmpo 7294 . 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 38098 . . 3 (𝐺𝑇 → (𝑍𝐺) = (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))))
3332adantl 485 . 2 ((𝐷𝑇𝐺𝑇) → (𝑍𝐺) = (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝐺)) ((𝑄𝑃) (𝑅‘(𝐺𝐷))))))
3422, 33eqtr4d 2860 1 ((𝐷𝑇𝐺𝑇) → (𝐷𝑌𝐺) = (𝑍𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114   ↦ cmpt 5122  ◡ccnv 5531   ∘ ccom 5536  ‘cfv 6334  ℩crio 7097  (class class class)co 7140   ∈ cmpo 7142  Basecbs 16474  lecple 16563  joincjn 17545  meetcmee 17546  Atomscatm 36517  LHypclh 37238  LTrncltrn 37355  trLctrl 37412 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145 This theorem is referenced by:  cdlemk31  38150  cdlemkuel-3  38152  cdlemkuv2-3N  38153  cdlemk18-3N  38154  cdlemk22-3  38155  cdlemkyu  38181
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