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Theorem cantnfsuc 8450
 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 7439 . . 3 (𝐾 ∈ ω → (𝐻‘suc 𝐾) = (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))(𝐻𝐾)))
32adantl 481 . 2 ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))(𝐻𝐾)))
4 elex 3185 . . . 4 (𝐾 ∈ ω → 𝐾 ∈ V)
54adantl 481 . . 3 ((𝜑𝐾 ∈ ω) → 𝐾 ∈ V)
6 fvex 6113 . . 3 (𝐻𝐾) ∈ V
7 simpl 472 . . . . . . . 8 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → 𝑢 = 𝐾)
87fveq2d 6107 . . . . . . 7 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐺𝑢) = (𝐺𝐾))
98oveq2d 6565 . . . . . 6 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐴𝑜 (𝐺𝑢)) = (𝐴𝑜 (𝐺𝐾)))
108fveq2d 6107 . . . . . 6 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐹‘(𝐺𝑢)) = (𝐹‘(𝐺𝐾)))
119, 10oveq12d 6567 . . . . 5 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → ((𝐴𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) = ((𝐴𝑜 (𝐺𝐾)) ·𝑜 (𝐹‘(𝐺𝐾))))
12 simpr 476 . . . . 5 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → 𝑣 = (𝐻𝐾))
1311, 12oveq12d 6567 . . . 4 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (((𝐴𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑣) = (((𝐴𝑜 (𝐺𝐾)) ·𝑜 (𝐹‘(𝐺𝐾))) +𝑜 (𝐻𝐾)))
14 fveq2 6103 . . . . . . . 8 (𝑘 = 𝑢 → (𝐺𝑘) = (𝐺𝑢))
1514oveq2d 6565 . . . . . . 7 (𝑘 = 𝑢 → (𝐴𝑜 (𝐺𝑘)) = (𝐴𝑜 (𝐺𝑢)))
1614fveq2d 6107 . . . . . . 7 (𝑘 = 𝑢 → (𝐹‘(𝐺𝑘)) = (𝐹‘(𝐺𝑢)))
1715, 16oveq12d 6567 . . . . . 6 (𝑘 = 𝑢 → ((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) = ((𝐴𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))))
1817oveq1d 6564 . . . . 5 (𝑘 = 𝑢 → (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧) = (((𝐴𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑧))
19 oveq2 6557 . . . . 5 (𝑧 = 𝑣 → (((𝐴𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑧) = (((𝐴𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑣))
2018, 19cbvmpt2v 6633 . . . 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 6577 . . . 4 (((𝐴𝑜 (𝐺𝐾)) ·𝑜 (𝐹‘(𝐺𝐾))) +𝑜 (𝐻𝐾)) ∈ V
2213, 20, 21ovmpt2a 6689 . . 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 693 . 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 2644 1 ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴𝑜 (𝐺𝐾)) ·𝑜 (𝐹‘(𝐺𝐾))) +𝑜 (𝐻𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874   E cep 4947  dom cdm 5038  Oncon0 5640  suc csuc 5642  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   supp csupp 7182  seq𝜔cseqom 7429   +𝑜 coa 7444   ·𝑜 comu 7445   ↑𝑜 coe 7446  OrdIsocoi 8297   CNF ccnf 8441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430 This theorem is referenced by:  cantnfle  8451  cantnflt  8452  cantnfp1lem3  8460  cantnflem1d  8468  cantnflem1  8469  cnfcomlem  8479
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