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Theorem cdlemksv 38020
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 6643 . . . . . 6 (𝑓 = 𝐺 → (𝑅𝑓) = (𝑅𝐺))
21oveq2d 7146 . . . . 5 (𝑓 = 𝐺 → (𝑃 (𝑅𝑓)) = (𝑃 (𝑅𝐺)))
3 coeq1 5701 . . . . . . 7 (𝑓 = 𝐺 → (𝑓𝐹) = (𝐺𝐹))
43fveq2d 6647 . . . . . 6 (𝑓 = 𝐺 → (𝑅‘(𝑓𝐹)) = (𝑅‘(𝐺𝐹)))
54oveq2d 7146 . . . . 5 (𝑓 = 𝐺 → ((𝑁𝑃) (𝑅‘(𝑓𝐹))) = ((𝑁𝑃) (𝑅‘(𝐺𝐹))))
62, 5oveq12d 7148 . . . 4 (𝑓 = 𝐺 → ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹)))) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹)))))
76eqeq2d 2832 . . 3 (𝑓 = 𝐺 → ((𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹)))) ↔ (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))))
87riotabidv 7090 . 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 7092 . 2 (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))) ∈ V
118, 9, 10fvmpt 6741 1 (𝐺𝑇 → (𝑆𝐺) = (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cmpt 5119  ccnv 5527  ccom 5532  cfv 6328  crio 7087  (class class class)co 7130  Basecbs 16461  lecple 16550  joincjn 17532  meetcmee 17533  Atomscatm 36439  LHypclh 37160  LTrncltrn 37277  trLctrl 37334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-riota 7088  df-ov 7133
This theorem is referenced by:  cdlemksel  38021  cdlemksv2  38023  cdlemkuvN  38040  cdlemkuel  38041  cdlemkuv2  38043  cdlemkuv-2N  38059  cdlemkuu  38071
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