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Theorem cdlemksv 40357
Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013.)
Hypotheses
Ref Expression
cdlemk.b 𝐡 = (Baseβ€˜πΎ)
cdlemk.l ≀ = (leβ€˜πΎ)
cdlemk.j ∨ = (joinβ€˜πΎ)
cdlemk.a 𝐴 = (Atomsβ€˜πΎ)
cdlemk.h 𝐻 = (LHypβ€˜πΎ)
cdlemk.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
cdlemk.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
cdlemk.m ∧ = (meetβ€˜πΎ)
cdlemk.s 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))
Assertion
Ref Expression
cdlemksv (𝐺 ∈ 𝑇 β†’ (π‘†β€˜πΊ) = (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))))
Distinct variable groups:   ∧ ,𝑓   ∨ ,𝑓   𝑓,𝐹   𝑓,𝑖,𝐺   𝑓,𝑁   𝑃,𝑓   𝑅,𝑓   𝑇,𝑓   𝑓,π‘Š
Allowed substitution hints:   𝐴(𝑓,𝑖)   𝐡(𝑓,𝑖)   𝑃(𝑖)   𝑅(𝑖)   𝑆(𝑓,𝑖)   𝑇(𝑖)   𝐹(𝑖)   𝐻(𝑓,𝑖)   ∨ (𝑖)   𝐾(𝑓,𝑖)   ≀ (𝑓,𝑖)   ∧ (𝑖)   𝑁(𝑖)   π‘Š(𝑖)

Proof of Theorem cdlemksv
StepHypRef Expression
1 fveq2 6902 . . . . . 6 (𝑓 = 𝐺 β†’ (π‘…β€˜π‘“) = (π‘…β€˜πΊ))
21oveq2d 7442 . . . . 5 (𝑓 = 𝐺 β†’ (𝑃 ∨ (π‘…β€˜π‘“)) = (𝑃 ∨ (π‘…β€˜πΊ)))
3 coeq1 5864 . . . . . . 7 (𝑓 = 𝐺 β†’ (𝑓 ∘ ◑𝐹) = (𝐺 ∘ ◑𝐹))
43fveq2d 6906 . . . . . 6 (𝑓 = 𝐺 β†’ (π‘…β€˜(𝑓 ∘ ◑𝐹)) = (π‘…β€˜(𝐺 ∘ ◑𝐹)))
54oveq2d 7442 . . . . 5 (𝑓 = 𝐺 β†’ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))) = ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))
62, 5oveq12d 7444 . . . 4 (𝑓 = 𝐺 β†’ ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹)))) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))))
76eqeq2d 2739 . . 3 (𝑓 = 𝐺 β†’ ((π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹)))) ↔ (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))))
87riotabidv 7384 . 2 (𝑓 = 𝐺 β†’ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))) = (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))))
9 cdlemk.s . 2 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))
10 riotaex 7386 . 2 (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))) ∈ V
118, 9, 10fvmpt 7010 1 (𝐺 ∈ 𝑇 β†’ (π‘†β€˜πΊ) = (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5235  β—‘ccnv 5681   ∘ ccom 5686  β€˜cfv 6553  β„©crio 7381  (class class class)co 7426  Basecbs 17189  lecple 17249  joincjn 18312  meetcmee 18313  Atomscatm 38775  LHypclh 39497  LTrncltrn 39614  trLctrl 39671
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-riota 7382  df-ov 7429
This theorem is referenced by:  cdlemksel  40358  cdlemksv2  40360  cdlemkuvN  40377  cdlemkuel  40378  cdlemkuv2  40380  cdlemkuv-2N  40396  cdlemkuu  40408
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