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Theorem cantnfval2 8863
 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 (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
cantnfcl.g 𝐺 = OrdIso( E , (𝐹 supp ∅))
cantnfcl.f (𝜑𝐹𝑆)
cantnfval.h 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)
Assertion
Ref Expression
cantnfval2 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +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.
StepHypRef Expression
1 cantnfs.s . . 3 𝑆 = dom (𝐴 CNF 𝐵)
2 cantnfs.a . . 3 (𝜑𝐴 ∈ On)
3 cantnfs.b . . 3 (𝜑𝐵 ∈ On)
4 cantnfcl.g . . 3 𝐺 = OrdIso( E , (𝐹 supp ∅))
5 cantnfcl.f . . 3 (𝜑𝐹𝑆)
6 cantnfval.h . . 3 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)
71, 2, 3, 4, 5, 6cantnfval 8862 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
8 ssid 3841 . . 3 dom 𝐺 ⊆ dom 𝐺
91, 2, 3, 4, 5cantnfcl 8861 . . . . 5 (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
109simprd 491 . . . 4 (𝜑 → dom 𝐺 ∈ ω)
11 sseq1 3844 . . . . . . 7 (𝑢 = ∅ → (𝑢 ⊆ dom 𝐺 ↔ ∅ ⊆ dom 𝐺))
12 fveq2 6446 . . . . . . . . 9 (𝑢 = ∅ → (𝐻𝑢) = (𝐻‘∅))
13 0ex 5026 . . . . . . . . . 10 ∅ ∈ V
146seqom0g 7834 . . . . . . . . . 10 (∅ ∈ V → (𝐻‘∅) = ∅)
1513, 14ax-mp 5 . . . . . . . . 9 (𝐻‘∅) = ∅
1612, 15syl6eq 2829 . . . . . . . 8 (𝑢 = ∅ → (𝐻𝑢) = ∅)
17 fveq2 6446 . . . . . . . . 9 (𝑢 = ∅ → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘∅))
18 eqid 2777 . . . . . . . . . . 11 seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅) = seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)
1918seqom0g 7834 . . . . . . . . . 10 (∅ ∈ V → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘∅) = ∅)
2013, 19ax-mp 5 . . . . . . . . 9 (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘∅) = ∅
2117, 20syl6eq 2829 . . . . . . . 8 (𝑢 = ∅ → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢) = ∅)
2216, 21eqeq12d 2792 . . . . . . 7 (𝑢 = ∅ → ((𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢) ↔ ∅ = ∅))
2311, 22imbi12d 336 . . . . . 6 (𝑢 = ∅ → ((𝑢 ⊆ dom 𝐺 → (𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢)) ↔ (∅ ⊆ dom 𝐺 → ∅ = ∅)))
2423imbi2d 332 . . . . 5 (𝑢 = ∅ → ((𝜑 → (𝑢 ⊆ dom 𝐺 → (𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢))) ↔ (𝜑 → (∅ ⊆ dom 𝐺 → ∅ = ∅))))
25 sseq1 3844 . . . . . . 7 (𝑢 = 𝑣 → (𝑢 ⊆ dom 𝐺𝑣 ⊆ dom 𝐺))
26 fveq2 6446 . . . . . . . 8 (𝑢 = 𝑣 → (𝐻𝑢) = (𝐻𝑣))
27 fveq2 6446 . . . . . . . 8 (𝑢 = 𝑣 → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣))
2826, 27eqeq12d 2792 . . . . . . 7 (𝑢 = 𝑣 → ((𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢) ↔ (𝐻𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)))
2925, 28imbi12d 336 . . . . . 6 (𝑢 = 𝑣 → ((𝑢 ⊆ dom 𝐺 → (𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢)) ↔ (𝑣 ⊆ dom 𝐺 → (𝐻𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣))))
3029imbi2d 332 . . . . 5 (𝑢 = 𝑣 → ((𝜑 → (𝑢 ⊆ dom 𝐺 → (𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢))) ↔ (𝜑 → (𝑣 ⊆ dom 𝐺 → (𝐻𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)))))
31 sseq1 3844 . . . . . . 7 (𝑢 = suc 𝑣 → (𝑢 ⊆ dom 𝐺 ↔ suc 𝑣 ⊆ dom 𝐺))
32 fveq2 6446 . . . . . . . 8 (𝑢 = suc 𝑣 → (𝐻𝑢) = (𝐻‘suc 𝑣))
33 fveq2 6446 . . . . . . . 8 (𝑢 = suc 𝑣 → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣))
3432, 33eqeq12d 2792 . . . . . . 7 (𝑢 = suc 𝑣 → ((𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢) ↔ (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣)))
3531, 34imbi12d 336 . . . . . 6 (𝑢 = suc 𝑣 → ((𝑢 ⊆ dom 𝐺 → (𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢)) ↔ (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣))))
3635imbi2d 332 . . . . 5 (𝑢 = suc 𝑣 → ((𝜑 → (𝑢 ⊆ dom 𝐺 → (𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢))) ↔ (𝜑 → (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣)))))
37 sseq1 3844 . . . . . . 7 (𝑢 = dom 𝐺 → (𝑢 ⊆ dom 𝐺 ↔ dom 𝐺 ⊆ dom 𝐺))
38 fveq2 6446 . . . . . . . 8 (𝑢 = dom 𝐺 → (𝐻𝑢) = (𝐻‘dom 𝐺))
39 fveq2 6446 . . . . . . . 8 (𝑢 = dom 𝐺 → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘dom 𝐺))
4038, 39eqeq12d 2792 . . . . . . 7 (𝑢 = dom 𝐺 → ((𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢) ↔ (𝐻‘dom 𝐺) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘dom 𝐺)))
4137, 40imbi12d 336 . . . . . 6 (𝑢 = dom 𝐺 → ((𝑢 ⊆ dom 𝐺 → (𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢)) ↔ (dom 𝐺 ⊆ dom 𝐺 → (𝐻‘dom 𝐺) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘dom 𝐺))))
4241imbi2d 332 . . . . 5 (𝑢 = dom 𝐺 → ((𝜑 → (𝑢 ⊆ dom 𝐺 → (𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢))) ↔ (𝜑 → (dom 𝐺 ⊆ dom 𝐺 → (𝐻‘dom 𝐺) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘dom 𝐺)))))
43 eqid 2777 . . . . . 6 ∅ = ∅
44432a1i 12 . . . . 5 (𝜑 → (∅ ⊆ dom 𝐺 → ∅ = ∅))
45 sssucid 6053 . . . . . . . . . 10 𝑣 ⊆ suc 𝑣
46 sstr 3828 . . . . . . . . . 10 ((𝑣 ⊆ suc 𝑣 ∧ suc 𝑣 ⊆ dom 𝐺) → 𝑣 ⊆ dom 𝐺)
4745, 46mpan 680 . . . . . . . . 9 (suc 𝑣 ⊆ dom 𝐺𝑣 ⊆ dom 𝐺)
4847imim1i 63 . . . . . . . 8 ((𝑣 ⊆ dom 𝐺 → (𝐻𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)) → (suc 𝑣 ⊆ dom 𝐺 → (𝐻𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)))
49 oveq2 6930 . . . . . . . . . . 11 ((𝐻𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣) → (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝑣)) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)))
506seqomsuc 7835 . . . . . . . . . . . . 13 (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝑣)))
5150ad2antrl 718 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → (𝐻‘suc 𝑣) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝑣)))
5218seqomsuc 7835 . . . . . . . . . . . . . 14 (𝑣 ∈ ω → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣) = (𝑣(𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)))
5352ad2antrl 718 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣) = (𝑣(𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)))
54 ssv 3843 . . . . . . . . . . . . . . . 16 dom 𝐺 ⊆ V
55 ssv 3843 . . . . . . . . . . . . . . . 16 On ⊆ V
56 resmpt2 7035 . . . . . . . . . . . . . . . 16 ((dom 𝐺 ⊆ V ∧ On ⊆ V) → ((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)) ↾ (dom 𝐺 × On)) = (𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)))
5754, 55, 56mp2an 682 . . . . . . . . . . . . . . 15 ((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)) ↾ (dom 𝐺 × On)) = (𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))
5857oveqi 6935 . . . . . . . . . . . . . 14 (𝑣((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)) ↾ (dom 𝐺 × On))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)) = (𝑣(𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣))
59 simprr 763 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → suc 𝑣 ⊆ dom 𝐺)
60 vex 3400 . . . . . . . . . . . . . . . . . 18 𝑣 ∈ V
6160sucid 6055 . . . . . . . . . . . . . . . . 17 𝑣 ∈ suc 𝑣
6261a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → 𝑣 ∈ suc 𝑣)
6359, 62sseldd 3821 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → 𝑣 ∈ dom 𝐺)
6418cantnfvalf 8859 . . . . . . . . . . . . . . . . 17 seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅):ω⟶On
6564ffvelrni 6622 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣) ∈ On)
6665ad2antrl 718 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣) ∈ On)
67 ovres 7077 . . . . . . . . . . . . . . 15 ((𝑣 ∈ dom 𝐺 ∧ (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣) ∈ On) → (𝑣((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)) ↾ (dom 𝐺 × On))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)))
6863, 66, 67syl2anc 579 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → (𝑣((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)) ↾ (dom 𝐺 × On))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)))
6958, 68syl5eqr 2827 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → (𝑣(𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)))
7053, 69eqtrd 2813 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)))
7151, 70eqeq12d 2792 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → ((𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣) ↔ (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝑣)) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣))))
7249, 71syl5ibr 238 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → ((𝐻𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣) → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣)))
7372expr 450 . . . . . . . . 9 ((𝜑𝑣 ∈ ω) → (suc 𝑣 ⊆ dom 𝐺 → ((𝐻𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣) → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣))))
7473a2d 29 . . . . . . . 8 ((𝜑𝑣 ∈ ω) → ((suc 𝑣 ⊆ dom 𝐺 → (𝐻𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)) → (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣))))
7548, 74syl5 34 . . . . . . 7 ((𝜑𝑣 ∈ ω) → ((𝑣 ⊆ dom 𝐺 → (𝐻𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)) → (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣))))
7675expcom 404 . . . . . 6 (𝑣 ∈ ω → (𝜑 → ((𝑣 ⊆ dom 𝐺 → (𝐻𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣)) → (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣)))))
7776a2d 29 . . . . 5 (𝑣 ∈ ω → ((𝜑 → (𝑣 ⊆ dom 𝐺 → (𝐻𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣))) → (𝜑 → (suc 𝑣 ⊆ dom 𝐺 → (𝐻‘suc 𝑣) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘suc 𝑣)))))
7824, 30, 36, 42, 44, 77finds 7370 . . . 4 (dom 𝐺 ∈ ω → (𝜑 → (dom 𝐺 ⊆ dom 𝐺 → (𝐻‘dom 𝐺) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘dom 𝐺))))
7910, 78mpcom 38 . . 3 (𝜑 → (dom 𝐺 ⊆ dom 𝐺 → (𝐻‘dom 𝐺) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘dom 𝐺)))
808, 79mpi 20 . 2 (𝜑 → (𝐻‘dom 𝐺) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘dom 𝐺))
817, 80eqtrd 2813 1 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘dom 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106  Vcvv 3397   ⊆ wss 3791  ∅c0 4140   E cep 5265   We wwe 5313   × cxp 5353  dom cdm 5355   ↾ cres 5357  Oncon0 5976  suc csuc 5978  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  ωcom 7343   supp csupp 7576  seq𝜔cseqom 7825   +o coa 7840   ·o comu 7841   ↑o coe 7842  OrdIsocoi 8703   CNF ccnf 8855 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-seqom 7826  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-oi 8704  df-cnf 8856 This theorem is referenced by:  cantnfres  8871
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