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Theorem cantnfsuc 9428
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 8288 . . 3 (𝐾 ∈ ω → (𝐻‘suc 𝐾) = (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝐾)))
32adantl 482 . 2 ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝐾)))
4 elex 3450 . . . 4 (𝐾 ∈ ω → 𝐾 ∈ V)
54adantl 482 . . 3 ((𝜑𝐾 ∈ ω) → 𝐾 ∈ V)
6 fvex 6787 . . 3 (𝐻𝐾) ∈ V
7 simpl 483 . . . . . . . 8 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → 𝑢 = 𝐾)
87fveq2d 6778 . . . . . . 7 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐺𝑢) = (𝐺𝐾))
98oveq2d 7291 . . . . . 6 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐴o (𝐺𝑢)) = (𝐴o (𝐺𝐾)))
108fveq2d 6778 . . . . . 6 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐹‘(𝐺𝑢)) = (𝐹‘(𝐺𝐾)))
119, 10oveq12d 7293 . . . . 5 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → ((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) = ((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))))
12 simpr 485 . . . . 5 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → 𝑣 = (𝐻𝐾))
1311, 12oveq12d 7293 . . . 4 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑣) = (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)))
14 fveq2 6774 . . . . . . . 8 (𝑘 = 𝑢 → (𝐺𝑘) = (𝐺𝑢))
1514oveq2d 7291 . . . . . . 7 (𝑘 = 𝑢 → (𝐴o (𝐺𝑘)) = (𝐴o (𝐺𝑢)))
1614fveq2d 6778 . . . . . . 7 (𝑘 = 𝑢 → (𝐹‘(𝐺𝑘)) = (𝐹‘(𝐺𝑢)))
1715, 16oveq12d 7293 . . . . . 6 (𝑘 = 𝑢 → ((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) = ((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))))
1817oveq1d 7290 . . . . 5 (𝑘 = 𝑢 → (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧) = (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑧))
19 oveq2 7283 . . . . 5 (𝑧 = 𝑣 → (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑧) = (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑣))
2018, 19cbvmpov 7370 . . . 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 7308 . . . 4 (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)) ∈ V
2213, 20, 21ovmpoa 7428 . . 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 586 . 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 2778 1 ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  c0 4256   E cep 5494  dom cdm 5589  Oncon0 6266  suc csuc 6268  cfv 6433  (class class class)co 7275  cmpo 7277  ωcom 7712   supp csupp 7977  seqωcseqom 8278   +o coa 8294   ·o comu 8295  o coe 8296  OrdIsocoi 9268   CNF ccnf 9419
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seqom 8279
This theorem is referenced by:  cantnfle  9429  cantnflt  9430  cantnfp1lem3  9438  cantnflem1d  9446  cantnflem1  9447  cnfcomlem  9457
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