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Theorem cantnfsuc 9661
Description: The value of the recursive function 𝐻 at a successor. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
cantnfcl.g 𝐺 = OrdIso( E , (𝐹 supp ∅))
cantnfcl.f (𝜑𝐹𝑆)
cantnfval.h 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)
Assertion
Ref Expression
cantnfsuc ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)))
Distinct variable groups:   𝑧,𝑘,𝐵   𝐴,𝑘,𝑧   𝑘,𝐹,𝑧   𝑆,𝑘,𝑧   𝑘,𝐺,𝑧   𝑘,𝐾,𝑧   𝜑,𝑘,𝑧
Allowed substitution hints:   𝐻(𝑧,𝑘)

Proof of Theorem cantnfsuc
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfval.h . . . 4 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)
21seqomsuc 8452 . . 3 (𝐾 ∈ ω → (𝐻‘suc 𝐾) = (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝐾)))
32adantl 483 . 2 ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝐾)))
4 elex 3493 . . . 4 (𝐾 ∈ ω → 𝐾 ∈ V)
54adantl 483 . . 3 ((𝜑𝐾 ∈ ω) → 𝐾 ∈ V)
6 fvex 6901 . . 3 (𝐻𝐾) ∈ V
7 simpl 484 . . . . . . . 8 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → 𝑢 = 𝐾)
87fveq2d 6892 . . . . . . 7 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐺𝑢) = (𝐺𝐾))
98oveq2d 7420 . . . . . 6 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐴o (𝐺𝑢)) = (𝐴o (𝐺𝐾)))
108fveq2d 6892 . . . . . 6 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐹‘(𝐺𝑢)) = (𝐹‘(𝐺𝐾)))
119, 10oveq12d 7422 . . . . 5 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → ((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) = ((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))))
12 simpr 486 . . . . 5 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → 𝑣 = (𝐻𝐾))
1311, 12oveq12d 7422 . . . 4 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑣) = (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)))
14 fveq2 6888 . . . . . . . 8 (𝑘 = 𝑢 → (𝐺𝑘) = (𝐺𝑢))
1514oveq2d 7420 . . . . . . 7 (𝑘 = 𝑢 → (𝐴o (𝐺𝑘)) = (𝐴o (𝐺𝑢)))
1614fveq2d 6892 . . . . . . 7 (𝑘 = 𝑢 → (𝐹‘(𝐺𝑘)) = (𝐹‘(𝐺𝑢)))
1715, 16oveq12d 7422 . . . . . 6 (𝑘 = 𝑢 → ((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) = ((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))))
1817oveq1d 7419 . . . . 5 (𝑘 = 𝑢 → (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧) = (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑧))
19 oveq2 7412 . . . . 5 (𝑧 = 𝑣 → (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑧) = (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑣))
2018, 19cbvmpov 7499 . . . 4 (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)) = (𝑢 ∈ V, 𝑣 ∈ V ↦ (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑣))
21 ovex 7437 . . . 4 (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)) ∈ V
2213, 20, 21ovmpoa 7558 . . 3 ((𝐾 ∈ V ∧ (𝐻𝐾) ∈ V) → (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝐾)) = (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)))
235, 6, 22sylancl 587 . 2 ((𝜑𝐾 ∈ ω) → (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝐾)) = (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)))
243, 23eqtrd 2773 1 ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  c0 4321   E cep 5578  dom cdm 5675  Oncon0 6361  suc csuc 6363  cfv 6540  (class class class)co 7404  cmpo 7406  ωcom 7850   supp csupp 8141  seqωcseqom 8442   +o coa 8458   ·o comu 8459  o coe 8460  OrdIsocoi 9500   CNF ccnf 9652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-seqom 8443
This theorem is referenced by:  cantnfle  9662  cantnflt  9663  cantnfp1lem3  9671  cantnflem1d  9679  cantnflem1  9680  cnfcomlem  9690
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