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Theorem coeaddlem 24411
Description: Lemma for coeadd 24413 and dgradd 24429. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
coefv0.1 𝐴 = (coeff‘𝐹)
coeadd.2 𝐵 = (coeff‘𝐺)
coeadd.3 𝑀 = (deg‘𝐹)
coeadd.4 𝑁 = (deg‘𝐺)
Assertion
Ref Expression
coeaddlem ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹𝑓 + 𝐺)) = (𝐴𝑓 + 𝐵) ∧ (deg‘(𝐹𝑓 + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀)))

Proof of Theorem coeaddlem
Dummy variables 𝑘 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyaddcl 24382 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 + 𝐺) ∈ (Poly‘ℂ))
2 coeadd.4 . . . . . 6 𝑁 = (deg‘𝐺)
3 dgrcl 24395 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
42, 3syl5eqel 2910 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
54adantl 475 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
6 coeadd.3 . . . . . 6 𝑀 = (deg‘𝐹)
7 dgrcl 24395 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
86, 7syl5eqel 2910 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
98adantr 474 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
105, 9ifcld 4353 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
11 addcl 10341 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
1211adantl 475 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
13 coefv0.1 . . . . . 6 𝐴 = (coeff‘𝐹)
1413coef3 24394 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
1514adantr 474 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
16 coeadd.2 . . . . . 6 𝐵 = (coeff‘𝐺)
1716coef3 24394 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
1817adantl 475 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
19 nn0ex 11632 . . . . 5 0 ∈ V
2019a1i 11 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
21 inidm 4049 . . . 4 (ℕ0 ∩ ℕ0) = ℕ0
2212, 15, 18, 20, 20, 21off 7177 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴𝑓 + 𝐵):ℕ0⟶ℂ)
23 oveq12 6919 . . . . . . . . . 10 (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑘) + (𝐵𝑘)) = (0 + 0))
24 00id 10537 . . . . . . . . . 10 (0 + 0) = 0
2523, 24syl6eq 2877 . . . . . . . . 9 (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑘) + (𝐵𝑘)) = 0)
2615ffnd 6283 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴 Fn ℕ0)
2718ffnd 6283 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵 Fn ℕ0)
28 eqidd 2826 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) = (𝐴𝑘))
29 eqidd 2826 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐵𝑘) = (𝐵𝑘))
3026, 27, 20, 20, 21, 28, 29ofval 7171 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑓 + 𝐵)‘𝑘) = ((𝐴𝑘) + (𝐵𝑘)))
3130eqeq1d 2827 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑓 + 𝐵)‘𝑘) = 0 ↔ ((𝐴𝑘) + (𝐵𝑘)) = 0))
3225, 31syl5ibr 238 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑓 + 𝐵)‘𝑘) = 0))
3332necon3ad 3012 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → ¬ ((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0)))
34 neorian 3093 . . . . . . 7 (((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0) ↔ ¬ ((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0))
3533, 34syl6ibr 244 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → ((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0)))
3613, 6dgrub2 24397 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
3736adantr 474 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
38 plyco0 24354 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀)))
399, 15, 38syl2anc 579 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀)))
4037, 39mpbid 224 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
4140r19.21bi 3141 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
429adantr 474 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
4342nn0red 11686 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℝ)
445adantr 474 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
4544nn0red 11686 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℝ)
46 max1 12311 . . . . . . . . . 10 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
4743, 45, 46syl2anc 579 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
48 nn0re 11635 . . . . . . . . . . 11 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
4948adantl 475 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
5010adantr 474 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
5150nn0red 11686 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ)
52 letr 10457 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ) → ((𝑘𝑀𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5349, 43, 51, 52syl3anc 1494 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘𝑀𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5447, 53mpan2d 685 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝑘𝑀𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5541, 54syld 47 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5616, 2dgrub2 24397 . . . . . . . . . . 11 (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
5756adantl 475 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
58 plyco0 24354 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐵:ℕ0⟶ℂ) → ((𝐵 “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁)))
595, 18, 58syl2anc 579 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐵 “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁)))
6057, 59mpbid 224 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁))
6160r19.21bi 3141 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵𝑘) ≠ 0 → 𝑘𝑁))
62 max2 12313 . . . . . . . . . 10 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
6343, 45, 62syl2anc 579 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
64 letr 10457 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ) → ((𝑘𝑁𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6549, 45, 51, 64syl3anc 1494 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘𝑁𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6663, 65mpan2d 685 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝑘𝑁𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6761, 66syld 47 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6855, 67jaod 890 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6935, 68syld 47 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
7069ralrimiva 3175 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
71 plyco0 24354 . . . . 5 ((if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0 ∧ (𝐴𝑓 + 𝐵):ℕ0⟶ℂ) → (((𝐴𝑓 + 𝐵) “ (ℤ‘(if(𝑀𝑁, 𝑁, 𝑀) + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀))))
7210, 22, 71syl2anc 579 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (((𝐴𝑓 + 𝐵) “ (ℤ‘(if(𝑀𝑁, 𝑁, 𝑀) + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀))))
7370, 72mpbird 249 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴𝑓 + 𝐵) “ (ℤ‘(if(𝑀𝑁, 𝑁, 𝑀) + 1))) = {0})
74 simpl 476 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
75 simpr 479 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
7613, 6coeid 24400 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
7776adantr 474 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
7816, 2coeid 24400 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))
7978adantl 475 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))
8074, 75, 9, 5, 15, 18, 37, 57, 77, 79plyaddlem1 24375 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀))(((𝐴𝑓 + 𝐵)‘𝑘) · (𝑧𝑘))))
811, 10, 22, 73, 80coeeq 24389 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓 + 𝐺)) = (𝐴𝑓 + 𝐵))
82 elfznn0 12734 . . . 4 (𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0)
83 ffvelrn 6611 . . . 4 (((𝐴𝑓 + 𝐵):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑓 + 𝐵)‘𝑘) ∈ ℂ)
8422, 82, 83syl2an 589 . . 3 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀))) → ((𝐴𝑓 + 𝐵)‘𝑘) ∈ ℂ)
851, 10, 84, 80dgrle 24405 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹𝑓 + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))
8681, 85jca 507 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹𝑓 + 𝐺)) = (𝐴𝑓 + 𝐵) ∧ (deg‘(𝐹𝑓 + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 878   = wceq 1656  wcel 2164  wne 2999  wral 3117  Vcvv 3414  ifcif 4308  {csn 4399   class class class wbr 4875  cmpt 4954  cima 5349  wf 6123  cfv 6127  (class class class)co 6910  𝑓 cof 7160  cc 10257  cr 10258  0cc0 10259  1c1 10260   + caddc 10262   · cmul 10264  cle 10399  0cn0 11625  cuz 11975  ...cfz 12626  cexp 13161  Σcsu 14800  Polycply 24346  coeffccoe 24348  degcdgr 24349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336  ax-pre-sup 10337  ax-addf 10338
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-of 7162  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-sup 8623  df-inf 8624  df-oi 8691  df-card 9085  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-div 11017  df-nn 11358  df-2 11421  df-3 11422  df-n0 11626  df-z 11712  df-uz 11976  df-rp 12120  df-fz 12627  df-fzo 12768  df-fl 12895  df-seq 13103  df-exp 13162  df-hash 13418  df-cj 14223  df-re 14224  df-im 14225  df-sqrt 14359  df-abs 14360  df-clim 14603  df-rlim 14604  df-sum 14801  df-0p 23843  df-ply 24350  df-coe 24352  df-dgr 24353
This theorem is referenced by:  coeadd  24413  dgradd  24429
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