Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemkuv-2N Structured version   Visualization version   GIF version

Theorem cdlemkuv-2N 39392
Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma2 (p) function, given 𝑉. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemk2.b 𝐡 = (Baseβ€˜πΎ)
cdlemk2.l ≀ = (leβ€˜πΎ)
cdlemk2.j ∨ = (joinβ€˜πΎ)
cdlemk2.m ∧ = (meetβ€˜πΎ)
cdlemk2.a 𝐴 = (Atomsβ€˜πΎ)
cdlemk2.h 𝐻 = (LHypβ€˜πΎ)
cdlemk2.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
cdlemk2.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
cdlemk2.s 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))
cdlemk2.q 𝑄 = (π‘†β€˜πΆ)
cdlemk2.v 𝑉 = (𝑑 ∈ 𝑇 ↦ (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘‘)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑑 ∘ ◑𝐢))))))
Assertion
Ref Expression
cdlemkuv-2N (𝐺 ∈ 𝑇 β†’ (π‘‰β€˜πΊ) = (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐢))))))
Distinct variable groups:   𝑓,𝑖, ∧   ≀ ,𝑖   ∨ ,𝑓,𝑖   𝐴,𝑖   𝐢,𝑓,𝑖   𝑓,𝐹,𝑖   𝑖,𝐻   𝑖,𝐾   𝑓,𝑁,𝑖   𝑃,𝑓,𝑖   𝑅,𝑓,𝑖   𝑇,𝑓,𝑖   𝑓,π‘Š,𝑖   ∧ ,𝑑   ∨ ,𝑑   𝐢,𝑑   π‘˜,𝑑,𝐺   𝑄,𝑑   𝑃,𝑑   𝑅,𝑑   𝑇,𝑑   π‘Š,𝑑
Allowed substitution hints:   𝐴(𝑓,π‘˜,𝑑)   𝐡(𝑓,𝑖,π‘˜,𝑑)   𝐢(π‘˜)   𝑃(π‘˜)   𝑄(𝑓,𝑖,π‘˜)   𝑅(π‘˜)   𝑆(𝑓,𝑖,π‘˜,𝑑)   𝑇(π‘˜)   𝐹(π‘˜,𝑑)   𝐺(𝑓,𝑖)   𝐻(𝑓,π‘˜,𝑑)   ∨ (π‘˜)   𝐾(𝑓,π‘˜,𝑑)   ≀ (𝑓,π‘˜,𝑑)   ∧ (π‘˜)   𝑁(π‘˜,𝑑)   𝑉(𝑓,𝑖,π‘˜,𝑑)   π‘Š(π‘˜)

Proof of Theorem cdlemkuv-2N
StepHypRef Expression
1 cdlemk2.b . 2 𝐡 = (Baseβ€˜πΎ)
2 cdlemk2.l . 2 ≀ = (leβ€˜πΎ)
3 cdlemk2.j . 2 ∨ = (joinβ€˜πΎ)
4 cdlemk2.a . 2 𝐴 = (Atomsβ€˜πΎ)
5 cdlemk2.h . 2 𝐻 = (LHypβ€˜πΎ)
6 cdlemk2.t . 2 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
7 cdlemk2.r . 2 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
8 cdlemk2.m . 2 ∧ = (meetβ€˜πΎ)
9 cdlemk2.v . 2 𝑉 = (𝑑 ∈ 𝑇 ↦ (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘‘)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑑 ∘ ◑𝐢))))))
101, 2, 3, 4, 5, 6, 7, 8, 9cdlemksv 39353 1 (𝐺 ∈ 𝑇 β†’ (π‘‰β€˜πΊ) = (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐢))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5189  β—‘ccnv 5633   ∘ ccom 5638  β€˜cfv 6497  β„©crio 7313  (class class class)co 7358  Basecbs 17088  lecple 17145  joincjn 18205  meetcmee 18206  Atomscatm 37771  LHypclh 38493  LTrncltrn 38610  trLctrl 38667
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 5257  ax-nul 5264  ax-pr 5385
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-riota 7314  df-ov 7361
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator