Theorem cantnfval2 8566

Description: Alternate expression for the value of the Cantor normal form function. (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
cantnfval2	⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘dom 𝐺))

Distinct variable groups:   𝑧,𝑘,𝐵   𝐴,𝑘,𝑧   𝑘,𝐹,𝑧   𝑆,𝑘,𝑧   𝑘,𝐺,𝑧   𝜑,𝑘,𝑧

Allowed substitution hints:   𝐻(𝑧,𝑘)

Proof of Theorem cantnfval2

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfs.s	. . 3 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
2		cantnfs.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
3		cantnfs.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
4		cantnfcl.g	. . 3 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
5		cantnfcl.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
6		cantnfval.h	. . 3 ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)
7	1, 2, 3, 4, 5, 6	cantnfval 8565	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
8		ssid 3624	. . 3 ⊢ dom 𝐺 ⊆ dom 𝐺
9	1, 2, 3, 4, 5	cantnfcl 8564	. . . . 5 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
10	9	simprd 479	. . . 4 ⊢ (𝜑 → dom 𝐺 ∈ ω)
11		sseq1 3626	. . . . . . 7 ⊢ (𝑢 = ∅ → (𝑢 ⊆ dom 𝐺 ↔ ∅ ⊆ dom 𝐺))
12		fveq2 6191	. . . . . . . . 9 ⊢ (𝑢 = ∅ → (𝐻‘𝑢) = (𝐻‘∅))
13		0ex 4790	. . . . . . . . . 10 ⊢ ∅ ∈ V
14	6	seqom0g 7551	. . . . . . . . . 10 ⊢ (∅ ∈ V → (𝐻‘∅) = ∅)
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ (𝐻‘∅) = ∅
16	12, 15	syl6eq 2672	. . . . . . . 8 ⊢ (𝑢 = ∅ → (𝐻‘𝑢) = ∅)
17		fveq2 6191	. . . . . . . . 9 ⊢ (𝑢 = ∅ → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘∅))
18		eqid 2622	. . . . . . . . . . 11 ⊢ seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)
19	18	seqom0g 7551	. . . . . . . . . 10 ⊢ (∅ ∈ V → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘∅) = ∅)
20	13, 19	ax-mp 5	. . . . . . . . 9 ⊢ (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘∅) = ∅
21	17, 20	syl6eq 2672	. . . . . . . 8 ⊢ (𝑢 = ∅ → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢) = ∅)
22	16, 21	eqeq12d 2637	. . . . . . 7 ⊢ (𝑢 = ∅ → ((𝐻‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢) ↔ ∅ = ∅))
23	11, 22	imbi12d 334	. . . . . 6 ⊢ (𝑢 = ∅ → ((𝑢 ⊆ dom 𝐺 → (𝐻‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢)) ↔ (∅ ⊆ dom 𝐺 → ∅ = ∅)))
24	23	imbi2d 330	. . . . 5 ⊢ (𝑢 = ∅ → ((𝜑 → (𝑢 ⊆ dom 𝐺 → (𝐻‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢))) ↔ (𝜑 → (∅ ⊆ dom 𝐺 → ∅ = ∅))))
25		sseq1 3626	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (𝑢 ⊆ dom 𝐺 ↔ 𝑣 ⊆ dom 𝐺))
26		fveq2 6191	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (𝐻‘𝑢) = (𝐻‘𝑣))
27		fveq2 6191	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣))
28	26, 27	eqeq12d 2637	. . . . . . 7 ⊢ (𝑢 = 𝑣 → ((𝐻‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢) ↔ (𝐻‘𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)))
29	25, 28	imbi12d 334	. . . . . 6 ⊢ (𝑢 = 𝑣 → ((𝑢 ⊆ dom 𝐺 → (𝐻‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢)) ↔ (𝑣 ⊆ dom 𝐺 → (𝐻‘𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣))))
30	29	imbi2d 330	. . . . 5 ⊢ (𝑢 = 𝑣 → ((𝜑 → (𝑢 ⊆ dom 𝐺 → (𝐻‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢))) ↔ (𝜑 → (𝑣 ⊆ dom 𝐺 → (𝐻‘𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)))))
31		sseq1 3626	. . . . . . 7 ⊢ (𝑢 = suc 𝑣 → (𝑢 ⊆ dom 𝐺 ↔ suc 𝑣 ⊆ dom 𝐺))
32		fveq2 6191	. . . . . . . 8 ⊢ (𝑢 = suc 𝑣 → (𝐻‘𝑢) = (𝐻‘suc 𝑣))
33		fveq2 6191	. . . . . . . 8 ⊢ (𝑢 = suc 𝑣 → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣))
34	32, 33	eqeq12d 2637	. . . . . . 7 ⊢ (𝑢 = suc 𝑣 → ((𝐻‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢) ↔ (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣)))
35	31, 34	imbi12d 334	. . . . . 6 ⊢ (𝑢 = suc 𝑣 → ((𝑢 ⊆ dom 𝐺 → (𝐻‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢)) ↔ (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣))))
36	35	imbi2d 330	. . . . 5 ⊢ (𝑢 = suc 𝑣 → ((𝜑 → (𝑢 ⊆ dom 𝐺 → (𝐻‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢))) ↔ (𝜑 → (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣)))))
37		sseq1 3626	. . . . . . 7 ⊢ (𝑢 = dom 𝐺 → (𝑢 ⊆ dom 𝐺 ↔ dom 𝐺 ⊆ dom 𝐺))
38		fveq2 6191	. . . . . . . 8 ⊢ (𝑢 = dom 𝐺 → (𝐻‘𝑢) = (𝐻‘dom 𝐺))
39		fveq2 6191	. . . . . . . 8 ⊢ (𝑢 = dom 𝐺 → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘dom 𝐺))
40	38, 39	eqeq12d 2637	. . . . . . 7 ⊢ (𝑢 = dom 𝐺 → ((𝐻‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢) ↔ (𝐻‘dom 𝐺) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘dom 𝐺)))
41	37, 40	imbi12d 334	. . . . . 6 ⊢ (𝑢 = dom 𝐺 → ((𝑢 ⊆ dom 𝐺 → (𝐻‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢)) ↔ (dom 𝐺 ⊆ dom 𝐺 → (𝐻‘dom 𝐺) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘dom 𝐺))))
42	41	imbi2d 330	. . . . 5 ⊢ (𝑢 = dom 𝐺 → ((𝜑 → (𝑢 ⊆ dom 𝐺 → (𝐻‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑢))) ↔ (𝜑 → (dom 𝐺 ⊆ dom 𝐺 → (𝐻‘dom 𝐺) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘dom 𝐺)))))
43		eqid 2622	. . . . . 6 ⊢ ∅ = ∅
44	43	2a1i 12	. . . . 5 ⊢ (𝜑 → (∅ ⊆ dom 𝐺 → ∅ = ∅))
45		sssucid 5802	. . . . . . . . . 10 ⊢ 𝑣 ⊆ suc 𝑣
46		sstr 3611	. . . . . . . . . 10 ⊢ ((𝑣 ⊆ suc 𝑣 ∧ suc 𝑣 ⊆ dom 𝐺) → 𝑣 ⊆ dom 𝐺)
47	45, 46	mpan 706	. . . . . . . . 9 ⊢ (suc 𝑣 ⊆ dom 𝐺 → 𝑣 ⊆ dom 𝐺)
48	47	imim1i 63	. . . . . . . 8 ⊢ ((𝑣 ⊆ dom 𝐺 → (𝐻‘𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)) → (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)))
49		oveq2 6658	. . . . . . . . . . 11 ⊢ ((𝐻‘𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣) → (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(𝐻‘𝑣)) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)))
50	6	seqomsuc 7552	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(𝐻‘𝑣)))
51	50	ad2antrl 764	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → (𝐻‘suc 𝑣) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(𝐻‘𝑣)))
52	18	seqomsuc 7552	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ω → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣) = (𝑣(𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)))
53	52	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣) = (𝑣(𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)))
54		ssv 3625	. . . . . . . . . . . . . . . 16 ⊢ dom 𝐺 ⊆ V
55		ssv 3625	. . . . . . . . . . . . . . . 16 ⊢ On ⊆ V
56		resmpt2 6758	. . . . . . . . . . . . . . . 16 ⊢ ((dom 𝐺 ⊆ V ∧ On ⊆ V) → ((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)) ↾ (dom 𝐺 × On)) = (𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)))
57	54, 55, 56	mp2an 708	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)) ↾ (dom 𝐺 × On)) = (𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))
58	57	oveqi 6663	. . . . . . . . . . . . . 14 ⊢ (𝑣((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)) ↾ (dom 𝐺 × On))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)) = (𝑣(𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣))
59		simprr 796	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → suc 𝑣 ⊆ dom 𝐺)
60		vex 3203	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑣 ∈ V
61	60	sucid 5804	. . . . . . . . . . . . . . . . 17 ⊢ 𝑣 ∈ suc 𝑣
62	61	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → 𝑣 ∈ suc 𝑣)
63	59, 62	sseldd 3604	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → 𝑣 ∈ dom 𝐺)
64	18	cantnfvalf 8562	. . . . . . . . . . . . . . . . 17 ⊢ seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅):ω⟶On
65	64	ffvelrni 6358	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ ω → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣) ∈ On)
66	65	ad2antrl 764	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣) ∈ On)
67		ovres 6800	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ dom 𝐺 ∧ (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣) ∈ On) → (𝑣((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)) ↾ (dom 𝐺 × On))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)))
68	63, 66, 67	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → (𝑣((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)) ↾ (dom 𝐺 × On))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)))
69	58, 68	syl5eqr 2670	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → (𝑣(𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)))
70	53, 69	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)))
71	51, 70	eqeq12d 2637	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → ((𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣) ↔ (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(𝐻‘𝑣)) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣))))
72	49, 71	syl5ibr 236	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → ((𝐻‘𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣) → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣)))
73	72	expr 643	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → (suc 𝑣 ⊆ dom 𝐺 → ((𝐻‘𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣) → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣))))
74	73	a2d 29	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ((suc 𝑣 ⊆ dom 𝐺 → (𝐻‘𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)) → (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣))))
75	48, 74	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ ω) → ((𝑣 ⊆ dom 𝐺 → (𝐻‘𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)) → (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣))))
76	75	expcom 451	. . . . . 6 ⊢ (𝑣 ∈ ω → (𝜑 → ((𝑣 ⊆ dom 𝐺 → (𝐻‘𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣)) → (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣)))))
77	76	a2d 29	. . . . 5 ⊢ (𝑣 ∈ ω → ((𝜑 → (𝑣 ⊆ dom 𝐺 → (𝐻‘𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘𝑣))) → (𝜑 → (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘suc 𝑣)))))
78	24, 30, 36, 42, 44, 77	finds 7092	. . . 4 ⊢ (dom 𝐺 ∈ ω → (𝜑 → (dom 𝐺 ⊆ dom 𝐺 → (𝐻‘dom 𝐺) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘dom 𝐺))))
79	10, 78	mpcom 38	. . 3 ⊢ (𝜑 → (dom 𝐺 ⊆ dom 𝐺 → (𝐻‘dom 𝐺) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘dom 𝐺)))
80	8, 79	mpi 20	. 2 ⊢ (𝜑 → (𝐻‘dom 𝐺) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘dom 𝐺))
81	7, 80	eqtrd 2656	1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)‘dom 𝐺))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  ∅c0 3915   E cep 5028   We wwe 5072   × cxp 5112  dom cdm 5114   ↾ cres 5116  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-rep 4771  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-fal 1489  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-rmo 2920  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-cnf 8559

This theorem is referenced by:  cantnfres  8574