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Theorem cdlemksv 40205
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 6881 . . . . . 6 (𝑓 = 𝐺 → (𝑅𝑓) = (𝑅𝐺))
21oveq2d 7417 . . . . 5 (𝑓 = 𝐺 → (𝑃 (𝑅𝑓)) = (𝑃 (𝑅𝐺)))
3 coeq1 5847 . . . . . . 7 (𝑓 = 𝐺 → (𝑓𝐹) = (𝐺𝐹))
43fveq2d 6885 . . . . . 6 (𝑓 = 𝐺 → (𝑅‘(𝑓𝐹)) = (𝑅‘(𝐺𝐹)))
54oveq2d 7417 . . . . 5 (𝑓 = 𝐺 → ((𝑁𝑃) (𝑅‘(𝑓𝐹))) = ((𝑁𝑃) (𝑅‘(𝐺𝐹))))
62, 5oveq12d 7419 . . . 4 (𝑓 = 𝐺 → ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹)))) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹)))))
76eqeq2d 2735 . . 3 (𝑓 = 𝐺 → ((𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹)))) ↔ (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))))
87riotabidv 7359 . 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 7361 . 2 (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))) ∈ V
118, 9, 10fvmpt 6988 1 (𝐺𝑇 → (𝑆𝐺) = (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝐺)) ((𝑁𝑃) (𝑅‘(𝐺𝐹))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cmpt 5221  ccnv 5665  ccom 5670  cfv 6533  crio 7356  (class class class)co 7401  Basecbs 17143  lecple 17203  joincjn 18266  meetcmee 18267  Atomscatm 38623  LHypclh 39345  LTrncltrn 39462  trLctrl 39519
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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6485  df-fun 6535  df-fv 6541  df-riota 7357  df-ov 7404
This theorem is referenced by:  cdlemksel  40206  cdlemksv2  40208  cdlemkuvN  40225  cdlemkuel  40226  cdlemkuv2  40228  cdlemkuv-2N  40244  cdlemkuu  40256
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