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Theorem coeaddlem 26228
Description: Lemma for coeadd 26230 and dgradd 26247. (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 26199 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f + 𝐺) ∈ (Poly‘ℂ))
2 coeadd.4 . . . . . 6 𝑁 = (deg‘𝐺)
3 dgrcl 26212 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
42, 3eqeltrid 2829 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
54adantl 480 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
6 coeadd.3 . . . . . 6 𝑀 = (deg‘𝐹)
7 dgrcl 26212 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
86, 7eqeltrid 2829 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
98adantr 479 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
105, 9ifcld 4576 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
11 addcl 11222 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
1211adantl 480 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
13 coefv0.1 . . . . . 6 𝐴 = (coeff‘𝐹)
1413coef3 26211 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
1514adantr 479 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
16 coeadd.2 . . . . . 6 𝐵 = (coeff‘𝐺)
1716coef3 26211 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
1817adantl 480 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
19 nn0ex 12511 . . . . 5 0 ∈ V
2019a1i 11 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
21 inidm 4217 . . . 4 (ℕ0 ∩ ℕ0) = ℕ0
2212, 15, 18, 20, 20, 21off 7703 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴f + 𝐵):ℕ0⟶ℂ)
23 oveq12 7428 . . . . . . . . . 10 (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑘) + (𝐵𝑘)) = (0 + 0))
24 00id 11421 . . . . . . . . . 10 (0 + 0) = 0
2523, 24eqtrdi 2781 . . . . . . . . 9 (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑘) + (𝐵𝑘)) = 0)
2615ffnd 6724 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴 Fn ℕ0)
2718ffnd 6724 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵 Fn ℕ0)
28 eqidd 2726 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) = (𝐴𝑘))
29 eqidd 2726 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐵𝑘) = (𝐵𝑘))
3026, 27, 20, 20, 21, 28, 29ofval 7696 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴f + 𝐵)‘𝑘) = ((𝐴𝑘) + (𝐵𝑘)))
3130eqeq1d 2727 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴f + 𝐵)‘𝑘) = 0 ↔ ((𝐴𝑘) + (𝐵𝑘)) = 0))
3225, 31imbitrrid 245 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴f + 𝐵)‘𝑘) = 0))
3332necon3ad 2942 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴f + 𝐵)‘𝑘) ≠ 0 → ¬ ((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0)))
34 neorian 3026 . . . . . . 7 (((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0) ↔ ¬ ((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0))
3533, 34imbitrrdi 251 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴f + 𝐵)‘𝑘) ≠ 0 → ((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0)))
3613, 6dgrub2 26214 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
3736adantr 479 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
38 plyco0 26171 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀)))
399, 15, 38syl2anc 582 . . . . . . . . . 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 3238 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
429adantr 479 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
4342nn0red 12566 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℝ)
445adantr 479 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
4544nn0red 12566 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℝ)
46 max1 13199 . . . . . . . . . 10 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
4743, 45, 46syl2anc 582 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
48 nn0re 12514 . . . . . . . . . . 11 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
4948adantl 480 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
5010adantr 479 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
5150nn0red 12566 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ)
52 letr 11340 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ) → ((𝑘𝑀𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5349, 43, 51, 52syl3anc 1368 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘𝑀𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5447, 53mpan2d 692 . . . . . . . 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 26214 . . . . . . . . . . 11 (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
5756adantl 480 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
58 plyco0 26171 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐵:ℕ0⟶ℂ) → ((𝐵 “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁)))
595, 18, 58syl2anc 582 . . . . . . . . . 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 3238 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵𝑘) ≠ 0 → 𝑘𝑁))
62 max2 13201 . . . . . . . . . 10 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
6343, 45, 62syl2anc 582 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
64 letr 11340 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ) → ((𝑘𝑁𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6549, 45, 51, 64syl3anc 1368 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘𝑁𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6663, 65mpan2d 692 . . . . . . . 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 857 . . . . . 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 3135 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 (((𝐴f + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
71 plyco0 26171 . . . . 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 582 . . . 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 481 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
75 simpr 483 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
7613, 6coeid 26217 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
7776adantr 479 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
7816, 2coeid 26217 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))
7978adantl 480 . . . 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 26192 . . 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 26206 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹f + 𝐺)) = (𝐴f + 𝐵))
82 elfznn0 13629 . . . 4 (𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0)
83 ffvelcdm 7090 . . . 4 (((𝐴f + 𝐵):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴f + 𝐵)‘𝑘) ∈ ℂ)
8422, 82, 83syl2an 594 . . 3 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀))) → ((𝐴f + 𝐵)‘𝑘) ∈ ℂ)
851, 10, 84, 80dgrle 26222 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹f + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))
8681, 85jca 510 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 394  wo 845   = wceq 1533  wcel 2098  wne 2929  wral 3050  Vcvv 3461  ifcif 4530  {csn 4630   class class class wbr 5149  cmpt 5232  cima 5681  wf 6545  cfv 6549  (class class class)co 7419  f cof 7683  cc 11138  cr 11139  0cc0 11140  1c1 11141   + caddc 11143   · cmul 11145  cle 11281  0cn0 12505  cuz 12855  ...cfz 13519  cexp 14062  Σcsu 15668  Polycply 26163  coeffccoe 26165  degcdgr 26166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-rp 13010  df-fz 13520  df-fzo 13663  df-fl 13793  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-rlim 15469  df-sum 15669  df-0p 25643  df-ply 26167  df-coe 26169  df-dgr 26170
This theorem is referenced by:  coeadd  26230  dgradd  26247
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