Theorem cantnfsuc 9571

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	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
cantnfcl.g	⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
cantnfcl.f	⊢ (𝜑 → 𝐹 ∈ 𝑆)
cantnfval.h	⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)

Assertion

Ref	Expression
cantnfsuc	⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)))

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.

Step	Hyp	Ref	Expression
1		cantnfval.h	. . . 4 ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)
2	1	seqomsuc 8385	. . 3 ⊢ (𝐾 ∈ ω → (𝐻‘suc 𝐾) = (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧))(𝐻‘𝐾)))
3	2	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧))(𝐻‘𝐾)))
4		elex 3458	. . . 4 ⊢ (𝐾 ∈ ω → 𝐾 ∈ V)
5	4	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → 𝐾 ∈ V)
6		fvex 6844	. . 3 ⊢ (𝐻‘𝐾) ∈ V
7		simpl 482	. . . . . . . 8 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → 𝑢 = 𝐾)
8	7	fveq2d 6835	. . . . . . 7 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐺‘𝑢) = (𝐺‘𝐾))
9	8	oveq2d 7371	. . . . . 6 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐴 ↑o (𝐺‘𝑢)) = (𝐴 ↑o (𝐺‘𝐾)))
10	8	fveq2d 6835	. . . . . 6 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐹‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝐾)))
11	9, 10	oveq12d 7373	. . . . 5 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → ((𝐴 ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) = ((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))))
12		simpr 484	. . . . 5 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → 𝑣 = (𝐻‘𝐾))
13	11, 12	oveq12d 7373	. . . 4 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (((𝐴 ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑣) = (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)))
14		fveq2 6831	. . . . . . . 8 ⊢ (𝑘 = 𝑢 → (𝐺‘𝑘) = (𝐺‘𝑢))
15	14	oveq2d 7371	. . . . . . 7 ⊢ (𝑘 = 𝑢 → (𝐴 ↑o (𝐺‘𝑘)) = (𝐴 ↑o (𝐺‘𝑢)))
16	14	fveq2d 6835	. . . . . . 7 ⊢ (𝑘 = 𝑢 → (𝐹‘(𝐺‘𝑘)) = (𝐹‘(𝐺‘𝑢)))
17	15, 16	oveq12d 7373	. . . . . 6 ⊢ (𝑘 = 𝑢 → ((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) = ((𝐴 ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))))
18	17	oveq1d 7370	. . . . 5 ⊢ (𝑘 = 𝑢 → (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧) = (((𝐴 ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑧))
19		oveq2 7363	. . . . 5 ⊢ (𝑧 = 𝑣 → (((𝐴 ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑧) = (((𝐴 ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑣))
20	18, 19	cbvmpov 7450	. . . 4 ⊢ (𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)) = (𝑢 ∈ V, 𝑣 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) +o 𝑣))
21		ovex 7388	. . . 4 ⊢ (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)) ∈ V
22	13, 20, 21	ovmpoa 7510	. . 3 ⊢ ((𝐾 ∈ V ∧ (𝐻‘𝐾) ∈ V) → (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧))(𝐻‘𝐾)) = (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)))
23	5, 6, 22	sylancl 586	. 2 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐾(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧))(𝐻‘𝐾)) = (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)))
24	3, 23	eqtrd 2768	1 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∅c0 4282   E cep 5520  dom cdm 5621  Oncon0 6314  suc csuc 6316  ‘cfv 6489  (class class class)co 7355   ∈ cmpo 7357  ωcom 7805   supp csupp 8099  seq_ωcseqom 8375   +_o coa 8391   ·_o comu 8392   ↑_o coe 8393  OrdIsocoi 9406   CNF ccnf 9562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-seqom 8376

This theorem is referenced by:  cantnfle  9572  cantnflt  9573  cantnfp1lem3  9581  cantnflem1d  9589  cantnflem1  9590  cnfcomlem  9600