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Theorem cantnfsuc 9711
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 8498 . . 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 3500 . . . 4 (𝐾 ∈ ω → 𝐾 ∈ V)
54adantl 481 . . 3 ((𝜑𝐾 ∈ ω) → 𝐾 ∈ V)
6 fvex 6918 . . 3 (𝐻𝐾) ∈ V
7 simpl 482 . . . . . . . 8 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → 𝑢 = 𝐾)
87fveq2d 6909 . . . . . . 7 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐺𝑢) = (𝐺𝐾))
98oveq2d 7448 . . . . . 6 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐴o (𝐺𝑢)) = (𝐴o (𝐺𝐾)))
108fveq2d 6909 . . . . . 6 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐹‘(𝐺𝑢)) = (𝐹‘(𝐺𝐾)))
119, 10oveq12d 7450 . . . . 5 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → ((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) = ((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))))
12 simpr 484 . . . . 5 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → 𝑣 = (𝐻𝐾))
1311, 12oveq12d 7450 . . . 4 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑣) = (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)))
14 fveq2 6905 . . . . . . . 8 (𝑘 = 𝑢 → (𝐺𝑘) = (𝐺𝑢))
1514oveq2d 7448 . . . . . . 7 (𝑘 = 𝑢 → (𝐴o (𝐺𝑘)) = (𝐴o (𝐺𝑢)))
1614fveq2d 6909 . . . . . . 7 (𝑘 = 𝑢 → (𝐹‘(𝐺𝑘)) = (𝐹‘(𝐺𝑢)))
1715, 16oveq12d 7450 . . . . . 6 (𝑘 = 𝑢 → ((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) = ((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))))
1817oveq1d 7447 . . . . 5 (𝑘 = 𝑢 → (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧) = (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑧))
19 oveq2 7440 . . . . 5 (𝑧 = 𝑣 → (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑧) = (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑣))
2018, 19cbvmpov 7529 . . . 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 7465 . . . 4 (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)) ∈ V
2213, 20, 21ovmpoa 7589 . . 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 2776 1 ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  c0 4332   E cep 5582  dom cdm 5684  Oncon0 6383  suc csuc 6385  cfv 6560  (class class class)co 7432  cmpo 7434  ωcom 7888   supp csupp 8186  seqωcseqom 8488   +o coa 8504   ·o comu 8505  o coe 8506  OrdIsocoi 9550   CNF ccnf 9702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-seqom 8489
This theorem is referenced by:  cantnfle  9712  cantnflt  9713  cantnfp1lem3  9721  cantnflem1d  9729  cantnflem1  9730  cnfcomlem  9740
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