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 Description: Lemma for coeadd 23811 and dgradd 23827. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
coefv0.1 𝐴 = (coeff‘𝐹)
Assertion
Ref Expression
coeaddlem ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹𝑓 + 𝐺)) = (𝐴𝑓 + 𝐵) ∧ (deg‘(𝐹𝑓 + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀)))

Dummy variables 𝑘 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyaddcl 23780 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 + 𝐺) ∈ (Poly‘ℂ))
2 coeadd.4 . . . . . 6 𝑁 = (deg‘𝐺)
3 dgrcl 23793 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
42, 3syl5eqel 2692 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
54adantl 481 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
6 coeadd.3 . . . . . 6 𝑀 = (deg‘𝐹)
7 dgrcl 23793 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
86, 7syl5eqel 2692 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
98adantr 480 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
105, 9ifcld 4081 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
11 addcl 9897 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
1211adantl 481 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
13 coefv0.1 . . . . . 6 𝐴 = (coeff‘𝐹)
1413coef3 23792 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
1514adantr 480 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
16 coeadd.2 . . . . . 6 𝐵 = (coeff‘𝐺)
1716coef3 23792 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
1817adantl 481 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
19 nn0ex 11175 . . . . 5 0 ∈ V
2019a1i 11 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
21 inidm 3784 . . . 4 (ℕ0 ∩ ℕ0) = ℕ0
2212, 15, 18, 20, 20, 21off 6810 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴𝑓 + 𝐵):ℕ0⟶ℂ)
23 oveq12 6558 . . . . . . . . . 10 (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑘) + (𝐵𝑘)) = (0 + 0))
24 00id 10090 . . . . . . . . . 10 (0 + 0) = 0
2523, 24syl6eq 2660 . . . . . . . . 9 (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑘) + (𝐵𝑘)) = 0)
26 ffn 5958 . . . . . . . . . . . 12 (𝐴:ℕ0⟶ℂ → 𝐴 Fn ℕ0)
2715, 26syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴 Fn ℕ0)
28 ffn 5958 . . . . . . . . . . . 12 (𝐵:ℕ0⟶ℂ → 𝐵 Fn ℕ0)
2918, 28syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵 Fn ℕ0)
30 eqidd 2611 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) = (𝐴𝑘))
31 eqidd 2611 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝐵𝑘) = (𝐵𝑘))
3227, 29, 20, 20, 21, 30, 31ofval 6804 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑓 + 𝐵)‘𝑘) = ((𝐴𝑘) + (𝐵𝑘)))
3332eqeq1d 2612 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑓 + 𝐵)‘𝑘) = 0 ↔ ((𝐴𝑘) + (𝐵𝑘)) = 0))
3425, 33syl5ibr 235 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0) → ((𝐴𝑓 + 𝐵)‘𝑘) = 0))
3534necon3ad 2795 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → ¬ ((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0)))
36 neorian 2876 . . . . . . 7 (((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0) ↔ ¬ ((𝐴𝑘) = 0 ∧ (𝐵𝑘) = 0))
3735, 36syl6ibr 241 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → ((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0)))
3813, 6dgrub2 23795 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
3938adantr 480 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
40 plyco0 23752 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀)))
419, 15, 40syl2anc 691 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴 “ (ℤ‘(𝑀 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀)))
4239, 41mpbid 221 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
4342r19.21bi 2916 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
449adantr 480 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℕ0)
4544nn0red 11229 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ ℝ)
465adantr 480 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ0)
4746nn0red 11229 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℝ)
48 max1 11890 . . . . . . . . . 10 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
4945, 47, 48syl2anc 691 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀))
50 nn0re 11178 . . . . . . . . . . 11 (𝑘 ∈ ℕ0𝑘 ∈ ℝ)
5150adantl 481 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℝ)
5210adantr 480 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℕ0)
5352nn0red 11229 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ)
54 letr 10010 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ) → ((𝑘𝑀𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5551, 45, 53, 54syl3anc 1318 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘𝑀𝑀 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5649, 55mpan2d 706 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝑘𝑀𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5743, 56syld 46 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
5816, 2dgrub2 23795 . . . . . . . . . . 11 (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
5958adantl 481 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
60 plyco0 23752 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐵:ℕ0⟶ℂ) → ((𝐵 “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁)))
615, 18, 60syl2anc 691 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐵 “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁)))
6259, 61mpbid 221 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 ((𝐵𝑘) ≠ 0 → 𝑘𝑁))
6362r19.21bi 2916 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵𝑘) ≠ 0 → 𝑘𝑁))
64 max2 11892 . . . . . . . . . 10 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
6545, 47, 64syl2anc 691 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → 𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀))
66 letr 10010 . . . . . . . . . 10 ((𝑘 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ if(𝑀𝑁, 𝑁, 𝑀) ∈ ℝ) → ((𝑘𝑁𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6751, 47, 53, 66syl3anc 1318 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝑘𝑁𝑁 ≤ if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6865, 67mpan2d 706 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (𝑘𝑁𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
6963, 68syld 46 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → ((𝐵𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
7057, 69jaod 394 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑘) ≠ 0 ∨ (𝐵𝑘) ≠ 0) → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
7137, 70syld 46 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ℕ0) → (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
7271ralrimiva 2949 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑘 ∈ ℕ0 (((𝐴𝑓 + 𝐵)‘𝑘) ≠ 0 → 𝑘 ≤ if(𝑀𝑁, 𝑁, 𝑀)))
73 plyco0 23752 . . . . 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 691 . . . 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 246 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴𝑓 + 𝐵) “ (ℤ‘(if(𝑀𝑁, 𝑁, 𝑀) + 1))) = {0})
76 simpl 472 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
77 simpr 476 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
7813, 6coeid 23798 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
7978adantr 480 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
8016, 2coeid 23798 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))
8180adantl 481 . . . 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 23773 . . 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 23787 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓 + 𝐺)) = (𝐴𝑓 + 𝐵))
84 elfznn0 12302 . . . 4 (𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀)) → 𝑘 ∈ ℕ0)
85 ffvelrn 6265 . . . 4 (((𝐴𝑓 + 𝐵):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑓 + 𝐵)‘𝑘) ∈ ℂ)
8622, 84, 85syl2an 493 . . 3 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀))) → ((𝐴𝑓 + 𝐵)‘𝑘) ∈ ℂ)
871, 10, 86, 82dgrle 23803 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹𝑓 + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))
8883, 87jca 553 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 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   ≤ cle 9954  ℕ0cn0 11169  ℤ≥cuz 11563  ...cfz 12197  ↑cexp 12722  Σcsu 14264  Polycply 23744  coeffccoe 23746  degcdgr 23747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-coe 23750  df-dgr 23751 This theorem is referenced by:  coeadd  23811  dgradd  23827
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