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Theorem cdlemksv 35951
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 6178 . . . . . 6 (𝑓 = 𝐺 → (𝑅𝑓) = (𝑅𝐺))
21oveq2d 6651 . . . . 5 (𝑓 = 𝐺 → (𝑃 (𝑅𝑓)) = (𝑃 (𝑅𝐺)))
3 coeq1 5268 . . . . . . 7 (𝑓 = 𝐺 → (𝑓𝐹) = (𝐺𝐹))
43fveq2d 6182 . . . . . 6 (𝑓 = 𝐺 → (𝑅‘(𝑓𝐹)) = (𝑅‘(𝐺𝐹)))
54oveq2d 6651 . . . . 5 (𝑓 = 𝐺 → ((𝑁𝑃) (𝑅‘(𝑓𝐹))) = ((𝑁𝑃) (𝑅‘(𝐺𝐹))))
62, 5oveq12d 6653 . . . 4 (𝑓 = 𝐺 → ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹)))) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹)))))
76eqeq2d 2630 . . 3 (𝑓 = 𝐺 → ((𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹)))) ↔ (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))))
87riotabidv 6598 . 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 6600 . 2 (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))) ∈ V
118, 9, 10fvmpt 6269 1 (𝐺𝑇 → (𝑆𝐺) = (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  cmpt 4720  ccnv 5103  ccom 5108  cfv 5876  crio 6595  (class class class)co 6635  Basecbs 15838  lecple 15929  joincjn 16925  meetcmee 16926  Atomscatm 34369  LHypclh 35089  LTrncltrn 35206  trLctrl 35264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-riota 6596  df-ov 6638
This theorem is referenced by:  cdlemksel  35952  cdlemksv2  35954  cdlemkuvN  35971  cdlemkuel  35972  cdlemkuv2  35974  cdlemkuv-2N  35990  cdlemkuu  36002
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