Theorem cantnfval2 9116

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 9115	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
8		ssid 3937	. . 3 ⊢ dom 𝐺 ⊆ dom 𝐺
9	1, 2, 3, 4, 5	cantnfcl 9114	. . . . 5 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
10	9	simprd 499	. . . 4 ⊢ (𝜑 → dom 𝐺 ∈ ω)
11		sseq1 3940	. . . . . . 7 ⊢ (𝑢 = ∅ → (𝑢 ⊆ dom 𝐺 ↔ ∅ ⊆ dom 𝐺))
12		fveq2 6645	. . . . . . . . 9 ⊢ (𝑢 = ∅ → (𝐻‘𝑢) = (𝐻‘∅))
13		0ex 5175	. . . . . . . . . 10 ⊢ ∅ ∈ V
14	6	seqom0g 8075	. . . . . . . . . 10 ⊢ (∅ ∈ V → (𝐻‘∅) = ∅)
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ (𝐻‘∅) = ∅
16	12, 15	eqtrdi 2849	. . . . . . . 8 ⊢ (𝑢 = ∅ → (𝐻‘𝑢) = ∅)
17		fveq2 6645	. . . . . . . . 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 2798	. . . . . . . . . . 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 8075	. . . . . . . . . 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	eqtrdi 2849	. . . . . . . 8 ⊢ (𝑢 = ∅ → (seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)‘𝑢) = ∅)
22	16, 21	eqeq12d 2814	. . . . . . 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 348	. . . . . 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 344	. . . . 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 3940	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (𝑢 ⊆ dom 𝐺 ↔ 𝑣 ⊆ dom 𝐺))
26		fveq2 6645	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (𝐻‘𝑢) = (𝐻‘𝑣))
27		fveq2 6645	. . . . . . . 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 2814	. . . . . . 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 348	. . . . . 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 344	. . . . 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 3940	. . . . . . 7 ⊢ (𝑢 = suc 𝑣 → (𝑢 ⊆ dom 𝐺 ↔ suc 𝑣 ⊆ dom 𝐺))
32		fveq2 6645	. . . . . . . 8 ⊢ (𝑢 = suc 𝑣 → (𝐻‘𝑢) = (𝐻‘suc 𝑣))
33		fveq2 6645	. . . . . . . 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 2814	. . . . . . 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 348	. . . . . 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 344	. . . . 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 3940	. . . . . . 7 ⊢ (𝑢 = dom 𝐺 → (𝑢 ⊆ dom 𝐺 ↔ dom 𝐺 ⊆ dom 𝐺))
38		fveq2 6645	. . . . . . . 8 ⊢ (𝑢 = dom 𝐺 → (𝐻‘𝑢) = (𝐻‘dom 𝐺))
39		fveq2 6645	. . . . . . . 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 2814	. . . . . . 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 348	. . . . . 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 344	. . . . 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 2798	. . . . . 6 ⊢ ∅ = ∅
44	43	2a1i 12	. . . . 5 ⊢ (𝜑 → (∅ ⊆ dom 𝐺 → ∅ = ∅))
45		sssucid 6236	. . . . . . . . . 10 ⊢ 𝑣 ⊆ suc 𝑣
46		sstr 3923	. . . . . . . . . 10 ⊢ ((𝑣 ⊆ suc 𝑣 ∧ suc 𝑣 ⊆ dom 𝐺) → 𝑣 ⊆ dom 𝐺)
47	45, 46	mpan 689	. . . . . . . . 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 7143	. . . . . . . . . . 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 8076	. . . . . . . . . . . . 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 727	. . . . . . . . . . . 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 8076	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . . 16 ⊢ dom 𝐺 ⊆ V
55		ssv 3939	. . . . . . . . . . . . . . . 16 ⊢ On ⊆ V
56		resmpo 7251	. . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . 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 7148	. . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → suc 𝑣 ⊆ dom 𝐺)
60		vex 3444	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑣 ∈ V
61	60	sucid 6238	. . . . . . . . . . . . . . . . 17 ⊢ 𝑣 ∈ suc 𝑣
62	61	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → 𝑣 ∈ suc 𝑣)
63	59, 62	sseldd 3916	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → 𝑣 ∈ dom 𝐺)
64	18	cantnfvalf 9112	. . . . . . . . . . . . . . . . 17 ⊢ seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅):ω⟶On
65	64	ffvelrni 6827	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ ω → (seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)‘𝑣) ∈ On)
66	65	ad2antrl 727	. . . . . . . . . . . . . . 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 7294	. . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 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 2847	. . . . . . . . . . . . 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 2833	. . . . . . . . . . . 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 2814	. . . . . . . . . . 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 249	. . . . . . . . . 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 460	. . . . . . . . 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 417	. . . . . 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 7589	. . . 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 2833	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 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  ∅c0 4243   E cep 5429   We wwe 5477   × cxp 5517  dom cdm 5519   ↾ cres 5521  Oncon0 6159  suc csuc 6161  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  ωcom 7560   supp csupp 7813  seq_ωcseqom 8066   +_o coa 8082   ·_o comu 8083   ↑_o coe 8084  OrdIsocoi 8957   CNF ccnf 9108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-seqom 8067  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-oi 8958  df-cnf 9109

This theorem is referenced by:  cantnfres  9124