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Theorem cdlemkuvN 39373
Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma1 (p) function π‘ˆ. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemk1.b 𝐡 = (Baseβ€˜πΎ)
cdlemk1.l ≀ = (leβ€˜πΎ)
cdlemk1.j ∨ = (joinβ€˜πΎ)
cdlemk1.m ∧ = (meetβ€˜πΎ)
cdlemk1.a 𝐴 = (Atomsβ€˜πΎ)
cdlemk1.h 𝐻 = (LHypβ€˜πΎ)
cdlemk1.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
cdlemk1.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
cdlemk1.s 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))
cdlemk1.o 𝑂 = (π‘†β€˜π·)
cdlemk1.u π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))
Assertion
Ref Expression
cdlemkuvN (𝐺 ∈ 𝑇 β†’ (π‘ˆβ€˜πΊ) = (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷))))))
Distinct variable groups:   𝑓,𝑖, ∧   ≀ ,𝑖   ∨ ,𝑓,𝑖   𝐴,𝑖   𝐷,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   ∧ ,𝑒   ∨ ,𝑒   𝐷,𝑒   𝑒,𝑗,𝐺   𝑒,𝑂   𝑃,𝑒   𝑅,𝑒   𝑇,𝑒   𝑒,π‘Š
Allowed substitution hints:   𝐴(𝑒,𝑓,𝑗)   𝐡(𝑒,𝑓,𝑖,𝑗)   𝐷(𝑗)   𝑃(𝑗)   𝑅(𝑗)   𝑆(𝑒,𝑓,𝑖,𝑗)   𝑇(𝑗)   π‘ˆ(𝑒,𝑓,𝑖,𝑗)   𝐹(𝑒,𝑗)   𝐺(𝑓,𝑖)   𝐻(𝑒,𝑓,𝑗)   ∨ (𝑗)   𝐾(𝑒,𝑓,𝑗)   ≀ (𝑒,𝑓,𝑗)   ∧ (𝑗)   𝑁(𝑒,𝑗)   𝑂(𝑓,𝑖,𝑗)   π‘Š(𝑗)

Proof of Theorem cdlemkuvN
StepHypRef Expression
1 cdlemk1.b . 2 𝐡 = (Baseβ€˜πΎ)
2 cdlemk1.l . 2 ≀ = (leβ€˜πΎ)
3 cdlemk1.j . 2 ∨ = (joinβ€˜πΎ)
4 cdlemk1.a . 2 𝐴 = (Atomsβ€˜πΎ)
5 cdlemk1.h . 2 𝐻 = (LHypβ€˜πΎ)
6 cdlemk1.t . 2 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
7 cdlemk1.r . 2 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
8 cdlemk1.m . 2 ∧ = (meetβ€˜πΎ)
9 cdlemk1.u . 2 π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))
101, 2, 3, 4, 5, 6, 7, 8, 9cdlemksv 39353 1 (𝐺 ∈ 𝑇 β†’ (π‘ˆβ€˜πΊ) = (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷))))))
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)
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