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Theorem cantnfval2 8816
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 8815 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
8 ssid 3827 . . 3 dom 𝐺 ⊆ dom 𝐺
91, 2, 3, 4, 5cantnfcl 8814 . . . . 5 (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
109simprd 485 . . . 4 (𝜑 → dom 𝐺 ∈ ω)
11 sseq1 3830 . . . . . . 7 (𝑢 = ∅ → (𝑢 ⊆ dom 𝐺 ↔ ∅ ⊆ dom 𝐺))
12 fveq2 6411 . . . . . . . . 9 (𝑢 = ∅ → (𝐻𝑢) = (𝐻‘∅))
13 0ex 4991 . . . . . . . . . 10 ∅ ∈ V
146seqom0g 7790 . . . . . . . . . 10 (∅ ∈ V → (𝐻‘∅) = ∅)
1513, 14ax-mp 5 . . . . . . . . 9 (𝐻‘∅) = ∅
1612, 15syl6eq 2863 . . . . . . . 8 (𝑢 = ∅ → (𝐻𝑢) = ∅)
17 fveq2 6411 . . . . . . . . 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 2813 . . . . . . . . . . 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 7790 . . . . . . . . . 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 2863 . . . . . . . 8 (𝑢 = ∅ → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)‘𝑢) = ∅)
2216, 21eqeq12d 2828 . . . . . . 7 (𝑢 = ∅ → ((𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)‘𝑢) ↔ ∅ = ∅))
2311, 22imbi12d 335 . . . . . 6 (𝑢 = ∅ → ((𝑢 ⊆ dom 𝐺 → (𝐻𝑢) = (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)‘𝑢)) ↔ (∅ ⊆ dom 𝐺 → ∅ = ∅)))
2423imbi2d 331 . . . . 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 3830 . . . . . . 7 (𝑢 = 𝑣 → (𝑢 ⊆ dom 𝐺𝑣 ⊆ dom 𝐺))
26 fveq2 6411 . . . . . . . 8 (𝑢 = 𝑣 → (𝐻𝑢) = (𝐻𝑣))
27 fveq2 6411 . . . . . . . 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 2828 . . . . . . 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 335 . . . . . 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 331 . . . . 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 3830 . . . . . . 7 (𝑢 = suc 𝑣 → (𝑢 ⊆ dom 𝐺 ↔ suc 𝑣 ⊆ dom 𝐺))
32 fveq2 6411 . . . . . . . 8 (𝑢 = suc 𝑣 → (𝐻𝑢) = (𝐻‘suc 𝑣))
33 fveq2 6411 . . . . . . . 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 2828 . . . . . . 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 335 . . . . . 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 331 . . . . 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 3830 . . . . . . 7 (𝑢 = dom 𝐺 → (𝑢 ⊆ dom 𝐺 ↔ dom 𝐺 ⊆ dom 𝐺))
38 fveq2 6411 . . . . . . . 8 (𝑢 = dom 𝐺 → (𝐻𝑢) = (𝐻‘dom 𝐺))
39 fveq2 6411 . . . . . . . 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 2828 . . . . . . 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 335 . . . . . 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 331 . . . . 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 2813 . . . . . 6 ∅ = ∅
44432a1i 12 . . . . 5 (𝜑 → (∅ ⊆ dom 𝐺 → ∅ = ∅))
45 sssucid 6021 . . . . . . . . . 10 𝑣 ⊆ suc 𝑣
46 sstr 3813 . . . . . . . . . 10 ((𝑣 ⊆ suc 𝑣 ∧ suc 𝑣 ⊆ dom 𝐺) → 𝑣 ⊆ dom 𝐺)
4745, 46mpan 673 . . . . . . . . 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 6885 . . . . . . . . . . 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 7791 . . . . . . . . . . . . 13 (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))(𝐻𝑣)))
5150ad2antrl 710 . . . . . . . . . . . 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 7791 . . . . . . . . . . . . . 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 710 . . . . . . . . . . . . 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 3829 . . . . . . . . . . . . . . . 16 dom 𝐺 ⊆ V
55 ssv 3829 . . . . . . . . . . . . . . . 16 On ⊆ V
56 resmpt2 6991 . . . . . . . . . . . . . . . 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 675 . . . . . . . . . . . . . . 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 6890 . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → suc 𝑣 ⊆ dom 𝐺)
60 vex 3401 . . . . . . . . . . . . . . . . . 18 𝑣 ∈ V
6160sucid 6023 . . . . . . . . . . . . . . . . 17 𝑣 ∈ suc 𝑣
6261a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → 𝑣 ∈ suc 𝑣)
6359, 62sseldd 3806 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → 𝑣 ∈ dom 𝐺)
6418cantnfvalf 8812 . . . . . . . . . . . . . . . . 17 seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅):ω⟶On
6564ffvelrni 6583 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (seq𝜔((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)‘𝑣) ∈ On)
6665ad2antrl 710 . . . . . . . . . . . . . . 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 7033 . . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . 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 2847 . . . . . . . . . . . 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 2828 . . . . . . . . . . 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 237 . . . . . . . . . 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 446 . . . . . . . . 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 400 . . . . . 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 7325 . . . 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 2847 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 384   = wceq 1637  wcel 2157  Vcvv 3398  wss 3776  c0 4123   E cep 5230   We wwe 5276   × cxp 5316  dom cdm 5318  cres 5320  Oncon0 5943  suc csuc 5945  cfv 6104  (class class class)co 6877  cmpt2 6879  ωcom 7298   supp csupp 7532  seq𝜔cseqom 7781   +𝑜 coa 7796   ·𝑜 comu 7797  𝑜 coe 7798  OrdIsocoi 8656   CNF ccnf 8808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-se 5278  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-isom 6113  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-supp 7533  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-seqom 7782  df-oadd 7803  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fsupp 8518  df-oi 8657  df-cnf 8809
This theorem is referenced by:  cantnfres  8824
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