Theorem cantnfsuc 9642

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 8446	. . 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 487	. 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 3478	. . . 4 ⊢ (𝐾 ∈ ω → 𝐾 ∈ V)
5	4	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → 𝐾 ∈ V)
6		fvex 6898	. . 3 ⊢ (𝐻‘𝐾) ∈ V
7		simpl 488	. . . . . . . 8 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → 𝑢 = 𝐾)
8	7	fveq2d 6889	. . . . . . 7 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐺‘𝑢) = (𝐺‘𝐾))
9	8	oveq2d 7432	. . . . . 6 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐴 ↑o (𝐺‘𝑢)) = (𝐴 ↑o (𝐺‘𝐾)))
10	8	fveq2d 6889	. . . . . 6 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → (𝐹‘(𝐺‘𝑢)) = (𝐹‘(𝐺‘𝐾)))
11	9, 10	oveq12d 7434	. . . . 5 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → ((𝐴 ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))) = ((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))))
12		simpr 490	. . . . 5 ⊢ ((𝑢 = 𝐾 ∧ 𝑣 = (𝐻‘𝐾)) → 𝑣 = (𝐻‘𝐾))
13	11, 12	oveq12d 7434	. . . 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 6885	. . . . . . . 8 ⊢ (𝑘 = 𝑢 → (𝐺‘𝑘) = (𝐺‘𝑢))
15	14	oveq2d 7432	. . . . . . 7 ⊢ (𝑘 = 𝑢 → (𝐴 ↑o (𝐺‘𝑘)) = (𝐴 ↑o (𝐺‘𝑢)))
16	14	fveq2d 6889	. . . . . . 7 ⊢ (𝑘 = 𝑢 → (𝐹‘(𝐺‘𝑘)) = (𝐹‘(𝐺‘𝑢)))
17	15, 16	oveq12d 7434	. . . . . 6 ⊢ (𝑘 = 𝑢 → ((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) = ((𝐴 ↑o (𝐺‘𝑢)) ·o (𝐹‘(𝐺‘𝑢))))
18	17	oveq1d 7431	. . . . 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 7424	. . . . 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 7511	. . . 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 7449	. . . 4 ⊢ (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)) ∈ V
22	13, 20, 21	ovmpoa 7571	. . 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 598	. 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 2800	1 ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ∅c0 4286   E cep 5562  dom cdm 5663  Oncon0 6364  suc csuc 6366  ‘cfv 6540  (class class class)co 7416   ∈ cmpo 7418  ωcom 7864   supp csupp 8158  seq_ωcseqom 8436   +_o coa 8452   ·_o comu 8453   ↑_o coe 8454  OrdIsocoi 9474   CNF ccnf 9633

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-seqom 8437

This theorem is used by:  cantnfle  9643  cantnflt  9644  cantnfp1lem3  9652  cantnflem1d  9660  cantnflem1  9661  cnfcomlem  9671