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Theorem cantnfsuc 8511
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 7497 . . 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 3198 . . . 4 (𝐾 ∈ ω → 𝐾 ∈ V)
54adantl 482 . . 3 ((𝜑𝐾 ∈ ω) → 𝐾 ∈ V)
6 fvex 6158 . . 3 (𝐻𝐾) ∈ V
7 simpl 473 . . . . . . . 8 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → 𝑢 = 𝐾)
87fveq2d 6152 . . . . . . 7 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐺𝑢) = (𝐺𝐾))
98oveq2d 6620 . . . . . 6 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐴𝑜 (𝐺𝑢)) = (𝐴𝑜 (𝐺𝐾)))
108fveq2d 6152 . . . . . 6 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐹‘(𝐺𝑢)) = (𝐹‘(𝐺𝐾)))
119, 10oveq12d 6622 . . . . 5 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → ((𝐴𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) = ((𝐴𝑜 (𝐺𝐾)) ·𝑜 (𝐹‘(𝐺𝐾))))
12 simpr 477 . . . . 5 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → 𝑣 = (𝐻𝐾))
1311, 12oveq12d 6622 . . . 4 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (((𝐴𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑣) = (((𝐴𝑜 (𝐺𝐾)) ·𝑜 (𝐹‘(𝐺𝐾))) +𝑜 (𝐻𝐾)))
14 fveq2 6148 . . . . . . . 8 (𝑘 = 𝑢 → (𝐺𝑘) = (𝐺𝑢))
1514oveq2d 6620 . . . . . . 7 (𝑘 = 𝑢 → (𝐴𝑜 (𝐺𝑘)) = (𝐴𝑜 (𝐺𝑢)))
1614fveq2d 6152 . . . . . . 7 (𝑘 = 𝑢 → (𝐹‘(𝐺𝑘)) = (𝐹‘(𝐺𝑢)))
1715, 16oveq12d 6622 . . . . . 6 (𝑘 = 𝑢 → ((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) = ((𝐴𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))))
1817oveq1d 6619 . . . . 5 (𝑘 = 𝑢 → (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧) = (((𝐴𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑧))
19 oveq2 6612 . . . . 5 (𝑧 = 𝑣 → (((𝐴𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑧) = (((𝐴𝑜 (𝐺𝑢)) ·𝑜 (𝐹‘(𝐺𝑢))) +𝑜 𝑣))
2018, 19cbvmpt2v 6688 . . . 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 6632 . . . 4 (((𝐴𝑜 (𝐺𝐾)) ·𝑜 (𝐹‘(𝐺𝐾))) +𝑜 (𝐻𝐾)) ∈ V
2213, 20, 21ovmpt2a 6744 . . 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 2655 1 ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴𝑜 (𝐺𝐾)) ·𝑜 (𝐹‘(𝐺𝐾))) +𝑜 (𝐻𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  c0 3891   E cep 4983  dom cdm 5074  Oncon0 5682  suc csuc 5684  cfv 5847  (class class class)co 6604  cmpt2 6606  ωcom 7012   supp csupp 7240  seq𝜔cseqom 7487   +𝑜 coa 7502   ·𝑜 comu 7503  𝑜 coe 7504  OrdIsocoi 8358   CNF ccnf 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488
This theorem is referenced by:  cantnfle  8512  cantnflt  8513  cantnfp1lem3  8521  cantnflem1d  8529  cantnflem1  8530  cnfcomlem  8540
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