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Theorem coeaddlem 25421
Description: Lemma for coeadd 25423 and dgradd 25439. (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 25392 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f + 𝐺) ∈ (Poly‘ℂ))
2 coeadd.4 . . . . . 6 𝑁 = (deg‘𝐺)
3 dgrcl 25405 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
42, 3eqeltrid 2845 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
54adantl 482 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
6 coeadd.3 . . . . . 6 𝑀 = (deg‘𝐹)
7 dgrcl 25405 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
86, 7eqeltrid 2845 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
98adantr 481 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
105, 9ifcld 4511 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
11 addcl 10964 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
1211adantl 482 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
13 coefv0.1 . . . . . 6 𝐴 = (coeff‘𝐹)
1413coef3 25404 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
1514adantr 481 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
16 coeadd.2 . . . . . 6 𝐵 = (coeff‘𝐺)
1716coef3 25404 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
1817adantl 482 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
19 nn0ex 12250 . . . . 5 0 ∈ V
2019a1i 11 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
21 inidm 4158 . . . 4 (ℕ0 ∩ ℕ0) = ℕ0
2212, 15, 18, 20, 20, 21off 7546 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴f + 𝐵):ℕ0⟶ℂ)
23 oveq12 7281 . . . . . . . . . 10 (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑘) + (𝐵𝑘)) = (0 + 0))
24 00id 11161 . . . . . . . . . 10 (0 + 0) = 0
2523, 24eqtrdi 2796 . . . . . . . . 9 (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑘) + (𝐵𝑘)) = 0)
2615ffnd 6599 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴 Fn ℕ0)
2718ffnd 6599 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵 Fn ℕ0)
28 eqidd 2741 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) = (𝐴𝑘))
29 eqidd 2741 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐵𝑘) = (𝐵𝑘))
3026, 27, 20, 20, 21, 28, 29ofval 7539 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴f + 𝐵)‘𝑘) = ((𝐴𝑘) + (𝐵𝑘)))
3130eqeq1d 2742 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴f + 𝐵)‘𝑘) = 0 ↔ ((𝐴𝑘) + (𝐵𝑘)) = 0))
3225, 31syl5ibr 245 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴f + 𝐵)‘𝑘) = 0))
3332necon3ad 2958 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴f + 𝐵)‘𝑘) ≠ 0 → ¬ ((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0)))
34 neorian 3041 . . . . . . 7 (((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0) ↔ ¬ ((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0))
3533, 34syl6ibr 251 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴f + 𝐵)‘𝑘) ≠ 0 → ((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0)))
3613, 6dgrub2 25407 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
3736adantr 481 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
38 plyco0 25364 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀)))
399, 15, 38syl2anc 584 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀)))
4037, 39mpbid 231 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
4140r19.21bi 3135 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
429adantr 481 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
4342nn0red 12305 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℝ)
445adantr 481 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
4544nn0red 12305 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℝ)
46 max1 12930 . . . . . . . . . 10 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
4743, 45, 46syl2anc 584 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
48 nn0re 12253 . . . . . . . . . . 11 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
4948adantl 482 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
5010adantr 481 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
5150nn0red 12305 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ)
52 letr 11080 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ) → ((𝑘𝑀𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5349, 43, 51, 52syl3anc 1370 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘𝑀𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5447, 53mpan2d 691 . . . . . . . 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 25407 . . . . . . . . . . 11 (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
5756adantl 482 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
58 plyco0 25364 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐵:ℕ0⟶ℂ) → ((𝐵 “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁)))
595, 18, 58syl2anc 584 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐵 “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁)))
6057, 59mpbid 231 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁))
6160r19.21bi 3135 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵𝑘) ≠ 0 → 𝑘𝑁))
62 max2 12932 . . . . . . . . . 10 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
6343, 45, 62syl2anc 584 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
64 letr 11080 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ) → ((𝑘𝑁𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6549, 45, 51, 64syl3anc 1370 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘𝑁𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6663, 65mpan2d 691 . . . . . . . 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 856 . . . . . 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 3110 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 (((𝐴f + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
71 plyco0 25364 . . . . 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 584 . . . 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 256 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴f + 𝐵) “ (ℤ‘(if(𝑀𝑁, 𝑁, 𝑀) + 1))) = {0})
74 simpl 483 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
75 simpr 485 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
7613, 6coeid 25410 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
7776adantr 481 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
7816, 2coeid 25410 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))
7978adantl 482 . . . 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 25385 . . 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 25399 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹f + 𝐺)) = (𝐴f + 𝐵))
82 elfznn0 13360 . . . 4 (𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0)
83 ffvelrn 6956 . . . 4 (((𝐴f + 𝐵):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴f + 𝐵)‘𝑘) ∈ ℂ)
8422, 82, 83syl2an 596 . . 3 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀))) → ((𝐴f + 𝐵)‘𝑘) ∈ ℂ)
851, 10, 84, 80dgrle 25415 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹f + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))
8681, 85jca 512 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 205  wa 396  wo 844   = wceq 1542  wcel 2110  wne 2945  wral 3066  Vcvv 3431  ifcif 4465  {csn 4567   class class class wbr 5079  cmpt 5162  cima 5593  wf 6428  cfv 6432  (class class class)co 7272  f cof 7526  cc 10880  cr 10881  0cc0 10882  1c1 10883   + caddc 10885   · cmul 10887  cle 11021  0cn0 12244  cuz 12593  ...cfz 13250  cexp 13793  Σcsu 15408  Polycply 25356  coeffccoe 25358  degcdgr 25359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7583  ax-inf2 9387  ax-cnex 10938  ax-resscn 10939  ax-1cn 10940  ax-icn 10941  ax-addcl 10942  ax-addrcl 10943  ax-mulcl 10944  ax-mulrcl 10945  ax-mulcom 10946  ax-addass 10947  ax-mulass 10948  ax-distr 10949  ax-i2m1 10950  ax-1ne0 10951  ax-1rid 10952  ax-rnegex 10953  ax-rrecex 10954  ax-cnre 10955  ax-pre-lttri 10956  ax-pre-lttrn 10957  ax-pre-ltadd 10958  ax-pre-mulgt0 10959  ax-pre-sup 10960
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-isom 6441  df-riota 7229  df-ov 7275  df-oprab 7276  df-mpo 7277  df-of 7528  df-om 7708  df-1st 7825  df-2nd 7826  df-frecs 8089  df-wrecs 8120  df-recs 8194  df-rdg 8233  df-1o 8289  df-er 8490  df-map 8609  df-pm 8610  df-en 8726  df-dom 8727  df-sdom 8728  df-fin 8729  df-sup 9189  df-inf 9190  df-oi 9257  df-card 9708  df-pnf 11022  df-mnf 11023  df-xr 11024  df-ltxr 11025  df-le 11026  df-sub 11218  df-neg 11219  df-div 11644  df-nn 11985  df-2 12047  df-3 12048  df-n0 12245  df-z 12331  df-uz 12594  df-rp 12742  df-fz 13251  df-fzo 13394  df-fl 13523  df-seq 13733  df-exp 13794  df-hash 14056  df-cj 14821  df-re 14822  df-im 14823  df-sqrt 14957  df-abs 14958  df-clim 15208  df-rlim 15209  df-sum 15409  df-0p 24845  df-ply 25360  df-coe 25362  df-dgr 25363
This theorem is referenced by:  coeadd  25423  dgradd  25439
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