Theorem cantnfsuc 8567

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 7552	. . 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 482	. 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 3212	. . . 4 ⊢ (𝐾 ∈ ω → 𝐾 ∈ V)
5	4	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → 𝐾 ∈ V)
6		fvex 6201	. . 3 ⊢ (𝐻‘𝐾) ∈ V
7		simpl 473	. . . . . . . 8 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → 𝑢 = 𝐾)
8	7	fveq2d 6195	. . . . . . 7 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐺‘𝑢) = (𝐺‘𝐾))
9	8	oveq2d 6666	. . . . . 6 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐴 ↑𝑜 (𝐺‘𝑢)) = (𝐴 ↑𝑜 (𝐺‘𝐾)))
10	8	fveq2d 6195	. . . . . 6 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐹‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝐾)))
11	9, 10	oveq12d 6668	. . . . 5 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → ((𝐴 ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))) = ((𝐴 ↑𝑜 (𝐺‘𝐾)) ·𝑜 (𝐹‘(𝐺‘𝐾))))
12		simpr 477	. . . . 5 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → 𝑣 = (𝐻‘𝐾))
13	11, 12	oveq12d 6668	. . . 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 6191	. . . . . . . 8 ⊢ (𝑘 = 𝑢 → (𝐺‘𝑘) = (𝐺‘𝑢))
15	14	oveq2d 6666	. . . . . . 7 ⊢ (𝑘 = 𝑢 → (𝐴 ↑𝑜 (𝐺‘𝑘)) = (𝐴 ↑𝑜 (𝐺‘𝑢)))
16	14	fveq2d 6195	. . . . . . 7 ⊢ (𝑘 = 𝑢 → (𝐹‘(𝐺‘𝑘)) = (𝐹‘(𝐺‘𝑢)))
17	15, 16	oveq12d 6668	. . . . . 6 ⊢ (𝑘 = 𝑢 → ((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) = ((𝐴 ↑𝑜 (𝐺‘𝑢)) ·𝑜 (𝐹‘(𝐺‘𝑢))))
18	17	oveq1d 6665	. . . . 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 6658	. . . . 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	cbvmpt2v 6735	. . . 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 6678	. . . 4 ⊢ (((𝐴 ↑𝑜 (𝐺‘𝐾)) ·𝑜 (𝐹‘(𝐺‘𝐾))) +𝑜 (𝐻‘𝐾)) ∈ V
22	13, 20, 21	ovmpt2a 6791	. . 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 694	. 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 2656	1 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴 ↑𝑜 (𝐺‘𝐾)) ·𝑜 (𝐹‘(𝐺‘𝐾))) +𝑜 (𝐻‘𝐾)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915   E cep 5028  dom cdm 5114  Oncon0 5723  suc csuc 5725  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065   supp csupp 7295  seq_𝜔cseqom 7542   +_𝑜 coa 7557   ·_𝑜 comu 7558   ↑_𝑜 coe 7559  OrdIsocoi 8414   CNF ccnf 8558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543

This theorem is referenced by:  cantnfle  8568  cantnflt  8569  cantnfp1lem3  8577  cantnflem1d  8585  cantnflem1  8586  cnfcomlem  8596