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Theorem cdlemkuu 39766
Description: Convert between function and operation forms of π‘Œ. TODO: Use operation form everywhere. (Contributed by NM, 6-Jul-2013.)
Hypotheses
Ref Expression
cdlemk3.b 𝐡 = (Baseβ€˜πΎ)
cdlemk3.l ≀ = (leβ€˜πΎ)
cdlemk3.j ∨ = (joinβ€˜πΎ)
cdlemk3.m ∧ = (meetβ€˜πΎ)
cdlemk3.a 𝐴 = (Atomsβ€˜πΎ)
cdlemk3.h 𝐻 = (LHypβ€˜πΎ)
cdlemk3.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
cdlemk3.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
cdlemk3.s 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))
cdlemk3.u1 π‘Œ = (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ (((π‘†β€˜π‘‘)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝑑))))))
cdlemk3.o2 𝑄 = (π‘†β€˜π·)
cdlemk3.u2 𝑍 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))
Assertion
Ref Expression
cdlemkuu ((𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (π·π‘ŒπΊ) = (π‘β€˜πΊ))
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   π‘Š,𝑑,𝑒,𝑓,𝑖
Allowed substitution hints:   𝐴(𝑒,𝑓,𝑗,𝑑)   𝐡(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)   π‘Š(𝑗)   π‘Œ(𝑒,𝑓,𝑖,𝑗,𝑑)   𝑍(𝑒,𝑓,𝑖,𝑗,𝑑)
Proof of Theorem cdlemkuu
StepHypRef Expression
1 fveq2 6892 . . . . . . . . 9 (𝑑 = 𝐷 β†’ (π‘†β€˜π‘‘) = (π‘†β€˜π·))
2 cdlemk3.o2 . . . . . . . . 9 𝑄 = (π‘†β€˜π·)
31, 2eqtr4di 2791 . . . . . . . 8 (𝑑 = 𝐷 β†’ (π‘†β€˜π‘‘) = 𝑄)
43fveq1d 6894 . . . . . . 7 (𝑑 = 𝐷 β†’ ((π‘†β€˜π‘‘)β€˜π‘ƒ) = (π‘„β€˜π‘ƒ))
5 cnveq 5874 . . . . . . . . 9 (𝑑 = 𝐷 β†’ ◑𝑑 = ◑𝐷)
65coeq2d 5863 . . . . . . . 8 (𝑑 = 𝐷 β†’ (𝑒 ∘ ◑𝑑) = (𝑒 ∘ ◑𝐷))
76fveq2d 6896 . . . . . . 7 (𝑑 = 𝐷 β†’ (π‘…β€˜(𝑒 ∘ ◑𝑑)) = (π‘…β€˜(𝑒 ∘ ◑𝐷)))
84, 7oveq12d 7427 . . . . . 6 (𝑑 = 𝐷 β†’ (((π‘†β€˜π‘‘)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝑑))) = ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))
98oveq2d 7425 . . . . 5 (𝑑 = 𝐷 β†’ ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ (((π‘†β€˜π‘‘)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝑑)))) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷)))))
109eqeq2d 2744 . . . 4 (𝑑 = 𝐷 β†’ ((π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ (((π‘†β€˜π‘‘)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝑑)))) ↔ (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))
1110riotabidv 7367 . . 3 (𝑑 = 𝐷 β†’ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ (((π‘†β€˜π‘‘)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝑑))))) = (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))
12 fveq2 6892 . . . . . . 7 (𝑒 = 𝐺 β†’ (π‘…β€˜π‘’) = (π‘…β€˜πΊ))
1312oveq2d 7425 . . . . . 6 (𝑒 = 𝐺 β†’ (𝑃 ∨ (π‘…β€˜π‘’)) = (𝑃 ∨ (π‘…β€˜πΊ)))
14 coeq1 5858 . . . . . . . 8 (𝑒 = 𝐺 β†’ (𝑒 ∘ ◑𝐷) = (𝐺 ∘ ◑𝐷))
1514fveq2d 6896 . . . . . . 7 (𝑒 = 𝐺 β†’ (π‘…β€˜(𝑒 ∘ ◑𝐷)) = (π‘…β€˜(𝐺 ∘ ◑𝐷)))
1615oveq2d 7425 . . . . . 6 (𝑒 = 𝐺 β†’ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))) = ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷))))
1713, 16oveq12d 7427 . . . . 5 (𝑒 = 𝐺 β†’ ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷)))) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷)))))
1817eqeq2d 2744 . . . 4 (𝑒 = 𝐺 β†’ ((π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷)))) ↔ (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷))))))
1918riotabidv 7367 . . 3 (𝑒 = 𝐺 β†’ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))) = (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷))))))
20 cdlemk3.u1 . . 3 π‘Œ = (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ (((π‘†β€˜π‘‘)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝑑))))))
21 riotaex 7369 . . 3 (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷))))) ∈ V
2211, 19, 20, 21ovmpo 7568 . 2 ((𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (π·π‘ŒπΊ) = (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷))))))
23 cdlemk3.b . . . 4 𝐡 = (Baseβ€˜πΎ)
24 cdlemk3.l . . . 4 ≀ = (leβ€˜πΎ)
25 cdlemk3.j . . . 4 ∨ = (joinβ€˜πΎ)
26 cdlemk3.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
27 cdlemk3.h . . . 4 𝐻 = (LHypβ€˜πΎ)
28 cdlemk3.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
29 cdlemk3.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
30 cdlemk3.m . . . 4 ∧ = (meetβ€˜πΎ)
31 cdlemk3.u2 . . . 4 𝑍 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))
3223, 24, 25, 26, 27, 28, 29, 30, 31cdlemksv 39715 . . 3 (𝐺 ∈ 𝑇 β†’ (π‘β€˜πΊ) = (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷))))))
3332adantl 483 . 2 ((𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (π‘β€˜πΊ) = (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷))))))
3422, 33eqtr4d 2776 1 ((𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (π·π‘ŒπΊ) = (π‘β€˜πΊ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5232  β—‘ccnv 5676   ∘ ccom 5681  β€˜cfv 6544  β„©crio 7364  (class class class)co 7409   ∈ cmpo 7411  Basecbs 17144  lecple 17204  joincjn 18264  meetcmee 18265  Atomscatm 38133  LHypclh 38855  LTrncltrn 38972  trLctrl 39029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414
This theorem is referenced by:  cdlemk31  39767  cdlemkuel-3  39769  cdlemkuv2-3N  39770  cdlemk18-3N  39771  cdlemk22-3  39772  cdlemkyu  39798
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