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Theorem cdlemkuv-2N 40267
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 40228 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 1533   ∈ wcel 2098   ↦ cmpt 5224  β—‘ccnv 5668   ∘ ccom 5673  β€˜cfv 6537  β„©crio 7360  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  meetcmee 18277  Atomscatm 38646  LHypclh 39368  LTrncltrn 39485  trLctrl 39542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-riota 7361  df-ov 7408
This theorem is referenced by: (None)
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