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Theorem plyco 14995
Description: The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
plyco.1 (𝜑𝐹 ∈ (Poly‘𝑆))
plyco.2 (𝜑𝐺 ∈ (Poly‘𝑆))
plyco.3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
plyco.4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
Assertion
Ref Expression
plyco (𝜑 → (𝐹𝐺) ∈ (Poly‘𝑆))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦

Proof of Theorem plyco
Dummy variables 𝑎 𝑘 𝑛 𝑧 𝑤 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyco.1 . . . 4 (𝜑𝐹 ∈ (Poly‘𝑆))
2 elply2 14971 . . . 4 (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))))
31, 2sylib 122 . . 3 (𝜑 → (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))))
43simprd 114 . 2 (𝜑 → ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘)))))
5 plyco.2 . . . . . . . . 9 (𝜑𝐺 ∈ (Poly‘𝑆))
6 plyf 14973 . . . . . . . . 9 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
75, 6syl 14 . . . . . . . 8 (𝜑𝐺:ℂ⟶ℂ)
87ffvelcdmda 5697 . . . . . . 7 ((𝜑𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
98ad4ant14 514 . . . . . 6 ((((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) ∧ 𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
107feqmptd 5614 . . . . . . 7 (𝜑𝐺 = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
1110ad2antrr 488 . . . . . 6 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
12 simprr 531 . . . . . 6 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))
13 oveq1 5929 . . . . . . . 8 (𝑥 = (𝐺𝑧) → (𝑥𝑘) = ((𝐺𝑧)↑𝑘))
1413oveq2d 5938 . . . . . . 7 (𝑥 = (𝐺𝑧) → ((𝑎𝑘) · (𝑥𝑘)) = ((𝑎𝑘) · ((𝐺𝑧)↑𝑘)))
1514sumeq2sdv 11535 . . . . . 6 (𝑥 = (𝐺𝑧) → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · ((𝐺𝑧)↑𝑘)))
169, 11, 12, 15fmptco 5728 . . . . 5 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) → (𝐹𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · ((𝐺𝑧)↑𝑘))))
17 oveq1 5929 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑥𝑘) = (𝑤𝑘))
1817oveq2d 5938 . . . . . . . . . . . 12 (𝑥 = 𝑤 → ((𝑎𝑘) · (𝑥𝑘)) = ((𝑎𝑘) · (𝑤𝑘)))
1918sumeq2sdv 11535 . . . . . . . . . . 11 (𝑥 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)))
2019cbvmptv 4129 . . . . . . . . . 10 (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)))
21 fveq2 5558 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝑎𝑘) = (𝑎𝑗))
22 oveq2 5930 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝑤𝑘) = (𝑤𝑗))
2321, 22oveq12d 5940 . . . . . . . . . . . 12 (𝑘 = 𝑗 → ((𝑎𝑘) · (𝑤𝑘)) = ((𝑎𝑗) · (𝑤𝑗)))
2423cbvsumv 11526 . . . . . . . . . . 11 Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))
2524mpteq2i 4120 . . . . . . . . . 10 (𝑤 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))
2620, 25eqtri 2217 . . . . . . . . 9 (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))
2726eqeq2i 2207 . . . . . . . 8 (𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))) ↔ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))))
2827anbi2i 457 . . . . . . 7 (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘)))) ↔ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))))
2928anbi2i 457 . . . . . 6 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) ↔ ((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))))))
301ad2antrr 488 . . . . . . 7 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))))) → 𝐹 ∈ (Poly‘𝑆))
315ad2antrr 488 . . . . . . 7 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))))) → 𝐺 ∈ (Poly‘𝑆))
32 plyco.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
3332ad4ant14 514 . . . . . . 7 ((((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
34 plyco.4 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
3534ad4ant14 514 . . . . . . 7 ((((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
36 simplrl 535 . . . . . . 7 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))))) → 𝑛 ∈ ℕ0)
37 simplrr 536 . . . . . . . . 9 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) → 𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))
383simpld 112 . . . . . . . . . . . . 13 (𝜑𝑆 ⊆ ℂ)
3938ad2antrr 488 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) → 𝑆 ⊆ ℂ)
40 cnex 8003 . . . . . . . . . . . 12 ℂ ∈ V
41 ssexg 4172 . . . . . . . . . . . 12 ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
4239, 40, 41sylancl 413 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) → 𝑆 ∈ V)
43 c0ex 8020 . . . . . . . . . . . 12 0 ∈ V
4443snex 4218 . . . . . . . . . . 11 {0} ∈ V
45 unexg 4478 . . . . . . . . . . 11 ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
4642, 44, 45sylancl 413 . . . . . . . . . 10 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) → (𝑆 ∪ {0}) ∈ V)
47 nn0ex 9255 . . . . . . . . . 10 0 ∈ V
48 elmapg 6720 . . . . . . . . . 10 (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ↔ 𝑎:ℕ0⟶(𝑆 ∪ {0})))
4946, 47, 48sylancl 413 . . . . . . . . 9 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ↔ 𝑎:ℕ0⟶(𝑆 ∪ {0})))
5037, 49mpbid 147 . . . . . . . 8 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
5129, 50sylbir 135 . . . . . . 7 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))))) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
52 simprl 529 . . . . . . 7 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))))) → (𝑎 “ (ℤ‘(𝑛 + 1))) = {0})
5329, 12sylbir 135 . . . . . . 7 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))))) → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))
5430, 31, 33, 35, 36, 51, 52, 53plycolemc 14994 . . . . . 6 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))))) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))
5529, 54sylbi 121 . . . . 5 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))
5616, 55eqeltrd 2273 . . . 4 (((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘))))) → (𝐹𝐺) ∈ (Poly‘𝑆))
5756ex 115 . . 3 ((𝜑 ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) → (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘)))) → (𝐹𝐺) ∈ (Poly‘𝑆)))
5857rexlimdvva 2622 . 2 (𝜑 → (∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑥𝑘)))) → (𝐹𝐺) ∈ (Poly‘𝑆)))
594, 58mpd 13 1 (𝜑 → (𝐹𝐺) ∈ (Poly‘𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2167  wrex 2476  Vcvv 2763  cun 3155  wss 3157  {csn 3622  cmpt 4094  cima 4666  ccom 4667  wf 5254  cfv 5258  (class class class)co 5922  𝑚 cmap 6707  cc 7877  0cc0 7879  1c1 7880   + caddc 7882   · cmul 7884  0cn0 9249  cuz 9601  ...cfz 10083  cexp 10630  Σcsu 11518  Polycply 14964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-disj 4011  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-of 6135  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-map 6709  df-en 6800  df-dom 6801  df-fin 6802  df-sup 7050  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519  df-ply 14966
This theorem is referenced by: (None)
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