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Theorem cdlemksv 35150
 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 𝐵 = (Base‘𝐾)
cdlemk.l = (le‘𝐾)
cdlemk.j = (join‘𝐾)
cdlemk.a 𝐴 = (Atoms‘𝐾)
cdlemk.h 𝐻 = (LHyp‘𝐾)
cdlemk.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk.r 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk.m = (meet‘𝐾)
cdlemk.s 𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))
Assertion
Ref Expression
cdlemksv (𝐺𝑇 → (𝑆𝐺) = (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))))
Distinct variable groups:   ,𝑓   ,𝑓   𝑓,𝐹   𝑓,𝑖,𝐺   𝑓,𝑁   𝑃,𝑓   𝑅,𝑓   𝑇,𝑓   𝑓,𝑊
Allowed substitution hints:   𝐴(𝑓,𝑖)   𝐵(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)   𝑊(𝑖)

Proof of Theorem cdlemksv
StepHypRef Expression
1 fveq2 6103 . . . . . 6 (𝑓 = 𝐺 → (𝑅𝑓) = (𝑅𝐺))
21oveq2d 6565 . . . . 5 (𝑓 = 𝐺 → (𝑃 (𝑅𝑓)) = (𝑃 (𝑅𝐺)))
3 coeq1 5201 . . . . . . 7 (𝑓 = 𝐺 → (𝑓𝐹) = (𝐺𝐹))
43fveq2d 6107 . . . . . 6 (𝑓 = 𝐺 → (𝑅‘(𝑓𝐹)) = (𝑅‘(𝐺𝐹)))
54oveq2d 6565 . . . . 5 (𝑓 = 𝐺 → ((𝑁𝑃) (𝑅‘(𝑓𝐹))) = ((𝑁𝑃) (𝑅‘(𝐺𝐹))))
62, 5oveq12d 6567 . . . 4 (𝑓 = 𝐺 → ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹)))) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹)))))
76eqeq2d 2620 . . 3 (𝑓 = 𝐺 → ((𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹)))) ↔ (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))))
87riotabidv 6513 . 2 (𝑓 = 𝐺 → (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))) = (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))))
9 cdlemk.s . 2 𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))
10 riotaex 6515 . 2 (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))) ∈ V
118, 9, 10fvmpt 6191 1 (𝐺𝑇 → (𝑆𝐺) = (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643  ◡ccnv 5037   ∘ ccom 5042  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  LHypclh 34288  LTrncltrn 34405  trLctrl 34463 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552 This theorem is referenced by:  cdlemksel  35151  cdlemksv2  35153  cdlemkuvN  35170  cdlemkuel  35171  cdlemkuv2  35173  cdlemkuv-2N  35189  cdlemkuu  35201
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