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Theorem cantnfsuc 9209
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 8125 . . 3 (𝐾 ∈ ω → (𝐻‘suc 𝐾) = (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝐾)))
32adantl 485 . 2 ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝐾)))
4 elex 3417 . . . 4 (𝐾 ∈ ω → 𝐾 ∈ V)
54adantl 485 . . 3 ((𝜑𝐾 ∈ ω) → 𝐾 ∈ V)
6 fvex 6690 . . 3 (𝐻𝐾) ∈ V
7 simpl 486 . . . . . . . 8 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → 𝑢 = 𝐾)
87fveq2d 6681 . . . . . . 7 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐺𝑢) = (𝐺𝐾))
98oveq2d 7189 . . . . . 6 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐴o (𝐺𝑢)) = (𝐴o (𝐺𝐾)))
108fveq2d 6681 . . . . . 6 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (𝐹‘(𝐺𝑢)) = (𝐹‘(𝐺𝐾)))
119, 10oveq12d 7191 . . . . 5 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → ((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) = ((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))))
12 simpr 488 . . . . 5 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → 𝑣 = (𝐻𝐾))
1311, 12oveq12d 7191 . . . 4 ((𝑢 = 𝐾𝑣 = (𝐻𝐾)) → (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑣) = (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)))
14 fveq2 6677 . . . . . . . 8 (𝑘 = 𝑢 → (𝐺𝑘) = (𝐺𝑢))
1514oveq2d 7189 . . . . . . 7 (𝑘 = 𝑢 → (𝐴o (𝐺𝑘)) = (𝐴o (𝐺𝑢)))
1614fveq2d 6681 . . . . . . 7 (𝑘 = 𝑢 → (𝐹‘(𝐺𝑘)) = (𝐹‘(𝐺𝑢)))
1715, 16oveq12d 7191 . . . . . 6 (𝑘 = 𝑢 → ((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) = ((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))))
1817oveq1d 7188 . . . . 5 (𝑘 = 𝑢 → (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧) = (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑧))
19 oveq2 7181 . . . . 5 (𝑧 = 𝑣 → (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑧) = (((𝐴o (𝐺𝑢)) ·o (𝐹‘(𝐺𝑢))) +o 𝑣))
2018, 19cbvmpov 7266 . . . 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 7206 . . . 4 (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)) ∈ V
2213, 20, 21ovmpoa 7323 . . 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 589 . 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 2774 1 ((𝜑𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴o (𝐺𝐾)) ·o (𝐹‘(𝐺𝐾))) +o (𝐻𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  Vcvv 3399  c0 4212   E cep 5434  dom cdm 5526  Oncon0 6173  suc csuc 6175  cfv 6340  (class class class)co 7173  cmpo 7175  ωcom 7602   supp csupp 7859  seqωcseqom 8115   +o coa 8131   ·o comu 8132  o coe 8133  OrdIsocoi 9049   CNF ccnf 9200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-om 7603  df-2nd 7718  df-wrecs 7979  df-recs 8040  df-rdg 8078  df-seqom 8116
This theorem is referenced by:  cantnfle  9210  cantnflt  9211  cantnfp1lem3  9219  cantnflem1d  9227  cantnflem1  9228  cnfcomlem  9238
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