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Theorem cantnfval2 9670
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 9669 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
8 ssid 4004 . . 3 dom 𝐺 ⊆ dom 𝐺
91, 2, 3, 4, 5cantnfcl 9668 . . . . 5 (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
109simprd 495 . . . 4 (𝜑 → dom 𝐺 ∈ ω)
11 sseq1 4007 . . . . . . 7 (𝑢 = ∅ → (𝑢 ⊆ dom 𝐺 ↔ ∅ ⊆ dom 𝐺))
12 fveq2 6891 . . . . . . . . 9 (𝑢 = ∅ → (𝐻𝑢) = (𝐻‘∅))
13 0ex 5307 . . . . . . . . . 10 ∅ ∈ V
146seqom0g 8462 . . . . . . . . . 10 (∅ ∈ V → (𝐻‘∅) = ∅)
1513, 14ax-mp 5 . . . . . . . . 9 (𝐻‘∅) = ∅
1612, 15eqtrdi 2787 . . . . . . . 8 (𝑢 = ∅ → (𝐻𝑢) = ∅)
17 fveq2 6891 . . . . . . . . 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 2731 . . . . . . . . . . 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 8462 . . . . . . . . . 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, 20eqtrdi 2787 . . . . . . . 8 (𝑢 = ∅ → (seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢) = ∅)
2216, 21eqeq12d 2747 . . . . . . 7 (𝑢 = ∅ → ((𝐻𝑢) = (seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢) ↔ ∅ = ∅))
2311, 22imbi12d 344 . . . . . 6 (𝑢 = ∅ → ((𝑢 ⊆ dom 𝐺 → (𝐻𝑢) = (seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑢)) ↔ (∅ ⊆ dom 𝐺 → ∅ = ∅)))
2423imbi2d 340 . . . . 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 4007 . . . . . . 7 (𝑢 = 𝑣 → (𝑢 ⊆ dom 𝐺𝑣 ⊆ dom 𝐺))
26 fveq2 6891 . . . . . . . 8 (𝑢 = 𝑣 → (𝐻𝑢) = (𝐻𝑣))
27 fveq2 6891 . . . . . . . 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 2747 . . . . . . 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 344 . . . . . 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 340 . . . . 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 4007 . . . . . . 7 (𝑢 = suc 𝑣 → (𝑢 ⊆ dom 𝐺 ↔ suc 𝑣 ⊆ dom 𝐺))
32 fveq2 6891 . . . . . . . 8 (𝑢 = suc 𝑣 → (𝐻𝑢) = (𝐻‘suc 𝑣))
33 fveq2 6891 . . . . . . . 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 2747 . . . . . . 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 344 . . . . . 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 340 . . . . 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 4007 . . . . . . 7 (𝑢 = dom 𝐺 → (𝑢 ⊆ dom 𝐺 ↔ dom 𝐺 ⊆ dom 𝐺))
38 fveq2 6891 . . . . . . . 8 (𝑢 = dom 𝐺 → (𝐻𝑢) = (𝐻‘dom 𝐺))
39 fveq2 6891 . . . . . . . 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 2747 . . . . . . 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 344 . . . . . 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 340 . . . . 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 2731 . . . . . 6 ∅ = ∅
44432a1i 12 . . . . 5 (𝜑 → (∅ ⊆ dom 𝐺 → ∅ = ∅))
45 sssucid 6444 . . . . . . . . . 10 𝑣 ⊆ suc 𝑣
46 sstr 3990 . . . . . . . . . 10 ((𝑣 ⊆ suc 𝑣 ∧ suc 𝑣 ⊆ dom 𝐺) → 𝑣 ⊆ dom 𝐺)
4745, 46mpan 687 . . . . . . . . 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 7420 . . . . . . . . . . 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 8463 . . . . . . . . . . . . 13 (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (𝑣(𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))(𝐻𝑣)))
5150ad2antrl 725 . . . . . . . . . . . 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 8463 . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . 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 4006 . . . . . . . . . . . . . . . 16 dom 𝐺 ⊆ V
55 ssv 4006 . . . . . . . . . . . . . . . 16 On ⊆ V
56 resmpo 7531 . . . . . . . . . . . . . . . 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 689 . . . . . . . . . . . . . . 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 7425 . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → suc 𝑣 ⊆ dom 𝐺)
60 vex 3477 . . . . . . . . . . . . . . . . . 18 𝑣 ∈ V
6160sucid 6446 . . . . . . . . . . . . . . . . 17 𝑣 ∈ suc 𝑣
6261a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → 𝑣 ∈ suc 𝑣)
6359, 62sseldd 3983 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑣 ∈ ω ∧ suc 𝑣 ⊆ dom 𝐺)) → 𝑣 ∈ dom 𝐺)
6418cantnfvalf 9666 . . . . . . . . . . . . . . . . 17 seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅):ω⟶On
6564ffvelcdmi 7085 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)‘𝑣) ∈ On)
6665ad2antrl 725 . . . . . . . . . . . . . . 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 7577 . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . 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, 68eqtr3id 2785 . . . . . . . . . . . . 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 2771 . . . . . . . . . . . 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 2747 . . . . . . . . . . 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, 71imbitrrid 245 . . . . . . . . . 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 456 . . . . . . . . 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 413 . . . . . 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 7893 . . . 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 2771 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 395   = wceq 1540  wcel 2105  Vcvv 3473  wss 3948  c0 4322   E cep 5579   We wwe 5630   × cxp 5674  dom cdm 5676  cres 5678  Oncon0 6364  suc csuc 6366  cfv 6543  (class class class)co 7412  cmpo 7414  ωcom 7859   supp csupp 8151  seqωcseqom 8453   +o coa 8469   ·o comu 8470  o coe 8471  OrdIsocoi 9510   CNF ccnf 9662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-seqom 8454  df-1o 8472  df-oadd 8476  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-oi 9511  df-cnf 9663
This theorem is referenced by:  cantnfres  9678
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