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Theorem coeaddlem 23922
Description: Lemma for coeadd 23924 and dgradd 23940. (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 23893 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 + 𝐺) ∈ (Poly‘ℂ))
2 coeadd.4 . . . . . 6 𝑁 = (deg‘𝐺)
3 dgrcl 23906 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
42, 3syl5eqel 2702 . . . . 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 23906 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
86, 7syl5eqel 2702 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
98adantr 481 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
105, 9ifcld 4108 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
11 addcl 9969 . . . . 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 23905 . . . . 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 23905 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
1817adantl 482 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
19 nn0ex 11249 . . . . 5 0 ∈ V
2019a1i 11 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
21 inidm 3805 . . . 4 (ℕ0 ∩ ℕ0) = ℕ0
2212, 15, 18, 20, 20, 21off 6872 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴𝑓 + 𝐵):ℕ0⟶ℂ)
23 oveq12 6619 . . . . . . . . . 10 (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑘) + (𝐵𝑘)) = (0 + 0))
24 00id 10162 . . . . . . . . . 10 (0 + 0) = 0
2523, 24syl6eq 2671 . . . . . . . . 9 (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑘) + (𝐵𝑘)) = 0)
26 ffn 6007 . . . . . . . . . . . 12 (𝐴:ℕ0⟶ℂ → 𝐴 Fn ℕ0)
2715, 26syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴 Fn ℕ0)
28 ffn 6007 . . . . . . . . . . . 12 (𝐵:ℕ0⟶ℂ → 𝐵 Fn ℕ0)
2918, 28syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵 Fn ℕ0)
30 eqidd 2622 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) = (𝐴𝑘))
31 eqidd 2622 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐵𝑘) = (𝐵𝑘))
3227, 29, 20, 20, 21, 30, 31ofval 6866 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑓 + 𝐵)‘𝑘) = ((𝐴𝑘) + (𝐵𝑘)))
3332eqeq1d 2623 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑓 + 𝐵)‘𝑘) = 0 ↔ ((𝐴𝑘) + (𝐵𝑘)) = 0))
3425, 33syl5ibr 236 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑓 + 𝐵)‘𝑘) = 0))
3534necon3ad 2803 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → ¬ ((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0)))
36 neorian 2884 . . . . . . 7 (((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0) ↔ ¬ ((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0))
3735, 36syl6ibr 242 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → ((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0)))
3813, 6dgrub2 23908 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
3938adantr 481 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
40 plyco0 23865 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀)))
419, 15, 40syl2anc 692 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀)))
4239, 41mpbid 222 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
4342r19.21bi 2927 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
449adantr 481 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
4544nn0red 11303 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℝ)
465adantr 481 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
4746nn0red 11303 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℝ)
48 max1 11966 . . . . . . . . . 10 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
4945, 47, 48syl2anc 692 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
50 nn0re 11252 . . . . . . . . . . 11 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
5150adantl 482 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
5210adantr 481 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
5352nn0red 11303 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ)
54 letr 10082 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ) → ((𝑘𝑀𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5551, 45, 53, 54syl3anc 1323 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘𝑀𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5649, 55mpan2d 709 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝑘𝑀𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5743, 56syld 47 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5816, 2dgrub2 23908 . . . . . . . . . . 11 (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
5958adantl 482 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
60 plyco0 23865 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐵:ℕ0⟶ℂ) → ((𝐵 “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁)))
615, 18, 60syl2anc 692 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐵 “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁)))
6259, 61mpbid 222 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁))
6362r19.21bi 2927 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵𝑘) ≠ 0 → 𝑘𝑁))
64 max2 11968 . . . . . . . . . 10 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
6545, 47, 64syl2anc 692 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
66 letr 10082 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ) → ((𝑘𝑁𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6751, 47, 53, 66syl3anc 1323 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘𝑁𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6865, 67mpan2d 709 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝑘𝑁𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6963, 68syld 47 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
7057, 69jaod 395 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
7137, 70syld 47 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
7271ralrimiva 2961 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
73 plyco0 23865 . . . . 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))))
7410, 22, 73syl2anc 692 . . . 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))))
7572, 74mpbird 247 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴𝑓 + 𝐵) “ (ℤ‘(if(𝑀𝑁, 𝑁, 𝑀) + 1))) = {0})
76 simpl 473 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
77 simpr 477 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
7813, 6coeid 23911 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
7978adantr 481 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
8016, 2coeid 23911 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))
8180adantl 482 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))
8276, 77, 9, 5, 15, 18, 39, 59, 79, 81plyaddlem1 23886 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀))(((𝐴𝑓 + 𝐵)‘𝑘) · (𝑧𝑘))))
831, 10, 22, 75, 82coeeq 23900 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓 + 𝐺)) = (𝐴𝑓 + 𝐵))
84 elfznn0 12381 . . . 4 (𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0)
85 ffvelrn 6318 . . . 4 (((𝐴𝑓 + 𝐵):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑓 + 𝐵)‘𝑘) ∈ ℂ)
8622, 84, 85syl2an 494 . . 3 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀))) → ((𝐴𝑓 + 𝐵)‘𝑘) ∈ ℂ)
871, 10, 86, 82dgrle 23916 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹𝑓 + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))
8883, 87jca 554 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 196  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3189  ifcif 4063  {csn 4153   class class class wbr 4618  cmpt 4678  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  𝑓 cof 6855  cc 9885  cr 9886  0cc0 9887  1c1 9888   + caddc 9890   · cmul 9892  cle 10026  0cn0 11243  cuz 11638  ...cfz 12275  cexp 12807  Σcsu 14357  Polycply 23857  coeffccoe 23859  degcdgr 23860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965  ax-addf 9966
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-inf 8300  df-oi 8366  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-z 11329  df-uz 11639  df-rp 11784  df-fz 12276  df-fzo 12414  df-fl 12540  df-seq 12749  df-exp 12808  df-hash 13065  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-clim 14160  df-rlim 14161  df-sum 14358  df-0p 23356  df-ply 23861  df-coe 23863  df-dgr 23864
This theorem is referenced by:  coeadd  23924  dgradd  23940
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