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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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