Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemkuu Structured version   Visualization version   GIF version
Theorem cdlemkuu 40069
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 6890 . . . . . . . . 9 (𝑑 = 𝐷 β†’ (π‘†β€˜π‘‘) = (π‘†β€˜π·))
2 cdlemk3.o2 . . . . . . . . 9 𝑄 = (π‘†β€˜π·)
31, 2eqtr4di 2788 . . . . . . . 8 (𝑑 = 𝐷 β†’ (π‘†β€˜π‘‘) = 𝑄)
43fveq1d 6892 . . . . . . 7 (𝑑 = 𝐷 β†’ ((π‘†β€˜π‘‘)β€˜π‘ƒ) = (π‘„β€˜π‘ƒ))
5 cnveq 5872 . . . . . . . . 9 (𝑑 = 𝐷 β†’ ◑𝑑 = ◑𝐷)
65coeq2d 5861 . . . . . . . 8 (𝑑 = 𝐷 β†’ (𝑒 ∘ ◑𝑑) = (𝑒 ∘ ◑𝐷))
76fveq2d 6894 . . . . . . 7 (𝑑 = 𝐷 β†’ (π‘…β€˜(𝑒 ∘ ◑𝑑)) = (π‘…β€˜(𝑒 ∘ ◑𝐷)))
84, 7oveq12d 7429 . . . . . 6 (𝑑 = 𝐷 β†’ (((π‘†β€˜π‘‘)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝑑))) = ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))
98oveq2d 7427 . . . . 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 2741 . . . 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 7369 . . 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 6890 . . . . . . 7 (𝑒 = 𝐺 β†’ (π‘…β€˜π‘’) = (π‘…β€˜πΊ))
1312oveq2d 7427 . . . . . 6 (𝑒 = 𝐺 β†’ (𝑃 ∨ (π‘…β€˜π‘’)) = (𝑃 ∨ (π‘…β€˜πΊ)))
14 coeq1 5856 . . . . . . . 8 (𝑒 = 𝐺 β†’ (𝑒 ∘ ◑𝐷) = (𝐺 ∘ ◑𝐷))
1514fveq2d 6894 . . . . . . 7 (𝑒 = 𝐺 β†’ (π‘…β€˜(𝑒 ∘ ◑𝐷)) = (π‘…β€˜(𝐺 ∘ ◑𝐷)))
1615oveq2d 7427 . . . . . 6 (𝑒 = 𝐺 β†’ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))) = ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷))))
1713, 16oveq12d 7429 . . . . 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 2741 . . . 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 7369 . . 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 7371 . . 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 7570 . 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 40018 . . 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 480 . 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 2773 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 394   = wceq 1539   ∈ wcel 2104   ↦ cmpt 5230  β—‘ccnv 5674   ∘ ccom 5679  β€˜cfv 6542  β„©crio 7366  (class class class)co 7411   ∈ cmpo 7413  Basecbs 17148  lecple 17208  joincjn 18268  meetcmee 18269  Atomscatm 38436  LHypclh 39158  LTrncltrn 39275  trLctrl 39332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416
This theorem is referenced by:  cdlemk31  40070  cdlemkuel-3  40072  cdlemkuv2-3N  40073  cdlemk18-3N  40074  cdlemk22-3  40075  cdlemkyu  40101
  Copyright terms: Public domain W3C validator