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Theorem coemulc 23932
 Description: The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
coemulc ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)) = ((ℕ0 × {𝐴}) ∘𝑓 · (coeff‘𝐹)))

Proof of Theorem coemulc
Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3608 . . . . 5 ℂ ⊆ ℂ
2 plyconst 23883 . . . . 5 ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
31, 2mpan 705 . . . 4 (𝐴 ∈ ℂ → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
4 plyssc 23877 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
54sseli 3583 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
6 plymulcl 23898 . . . 4 (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝐹 ∈ (Poly‘ℂ)) → ((ℂ × {𝐴}) ∘𝑓 · 𝐹) ∈ (Poly‘ℂ))
73, 5, 6syl2an 494 . . 3 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ × {𝐴}) ∘𝑓 · 𝐹) ∈ (Poly‘ℂ))
8 eqid 2621 . . . 4 (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)) = (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹))
98coef3 23909 . . 3 (((ℂ × {𝐴}) ∘𝑓 · 𝐹) ∈ (Poly‘ℂ) → (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)):ℕ0⟶ℂ)
10 ffn 6007 . . 3 ((coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)):ℕ0⟶ℂ → (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)) Fn ℕ0)
117, 9, 103syl 18 . 2 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)) Fn ℕ0)
12 fconstg 6054 . . . . 5 (𝐴 ∈ ℂ → (ℕ0 × {𝐴}):ℕ0⟶{𝐴})
1312adantr 481 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ0 × {𝐴}):ℕ0⟶{𝐴})
14 ffn 6007 . . . 4 ((ℕ0 × {𝐴}):ℕ0⟶{𝐴} → (ℕ0 × {𝐴}) Fn ℕ0)
1513, 14syl 17 . . 3 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ0 × {𝐴}) Fn ℕ0)
16 eqid 2621 . . . . . 6 (coeff‘𝐹) = (coeff‘𝐹)
1716coef3 23909 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
1817adantl 482 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
19 ffn 6007 . . . 4 ((coeff‘𝐹):ℕ0⟶ℂ → (coeff‘𝐹) Fn ℕ0)
2018, 19syl 17 . . 3 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹) Fn ℕ0)
21 nn0ex 11250 . . . 4 0 ∈ V
2221a1i 11 . . 3 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
23 inidm 3805 . . 3 (ℕ0 ∩ ℕ0) = ℕ0
2415, 20, 22, 22, 23offn 6868 . 2 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℕ0 × {𝐴}) ∘𝑓 · (coeff‘𝐹)) Fn ℕ0)
253ad2antrr 761 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26 eqid 2621 . . . . . . 7 (coeff‘(ℂ × {𝐴})) = (coeff‘(ℂ × {𝐴}))
2726coefv0 23925 . . . . . 6 ((ℂ × {𝐴}) ∈ (Poly‘ℂ) → ((ℂ × {𝐴})‘0) = ((coeff‘(ℂ × {𝐴}))‘0))
2825, 27syl 17 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ × {𝐴})‘0) = ((coeff‘(ℂ × {𝐴}))‘0))
29 simpll 789 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ ℂ)
30 0cn 9984 . . . . . 6 0 ∈ ℂ
31 fvconst2g 6427 . . . . . 6 ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ((ℂ × {𝐴})‘0) = 𝐴)
3229, 30, 31sylancl 693 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ × {𝐴})‘0) = 𝐴)
3328, 32eqtr3d 2657 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(ℂ × {𝐴}))‘0) = 𝐴)
34 simpr 477 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
3534nn0cnd 11305 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℂ)
3635subid1d 10333 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (𝑛 − 0) = 𝑛)
3736fveq2d 6157 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 0)) = ((coeff‘𝐹)‘𝑛))
3833, 37oveq12d 6628 . . 3 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) = (𝐴 · ((coeff‘𝐹)‘𝑛)))
395ad2antlr 762 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℂ))
4026, 16coemul 23929 . . . . 5 (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝐹 ∈ (Poly‘ℂ) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹))‘𝑛) = Σ𝑘 ∈ (0...𝑛)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))))
4125, 39, 34, 40syl3anc 1323 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹))‘𝑛) = Σ𝑘 ∈ (0...𝑛)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))))
42 nn0uz 11674 . . . . . . 7 0 = (ℤ‘0)
4334, 42syl6eleq 2708 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ (ℤ‘0))
44 fzss2 12331 . . . . . 6 (𝑛 ∈ (ℤ‘0) → (0...0) ⊆ (0...𝑛))
4543, 44syl 17 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...0) ⊆ (0...𝑛))
46 elfz1eq 12302 . . . . . . . 8 (𝑘 ∈ (0...0) → 𝑘 = 0)
4746adantl 482 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → 𝑘 = 0)
48 fveq2 6153 . . . . . . . 8 (𝑘 = 0 → ((coeff‘(ℂ × {𝐴}))‘𝑘) = ((coeff‘(ℂ × {𝐴}))‘0))
49 oveq2 6618 . . . . . . . . 9 (𝑘 = 0 → (𝑛𝑘) = (𝑛 − 0))
5049fveq2d 6157 . . . . . . . 8 (𝑘 = 0 → ((coeff‘𝐹)‘(𝑛𝑘)) = ((coeff‘𝐹)‘(𝑛 − 0)))
5148, 50oveq12d 6628 . . . . . . 7 (𝑘 = 0 → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
5247, 51syl 17 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
5318ffvelrnda 6320 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘𝑛) ∈ ℂ)
5429, 53mulcld 10012 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (𝐴 · ((coeff‘𝐹)‘𝑛)) ∈ ℂ)
5538, 54eqeltrd 2698 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ)
5655adantr 481 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ)
5752, 56eqeltrd 2698 . . . . 5 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))) ∈ ℂ)
58 eldifn 3716 . . . . . . . . 9 (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → ¬ 𝑘 ∈ (0...0))
5958adantl 482 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ¬ 𝑘 ∈ (0...0))
60 eldifi 3715 . . . . . . . . . . . . 13 (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ (0...𝑛))
61 elfznn0 12382 . . . . . . . . . . . . 13 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
6260, 61syl 17 . . . . . . . . . . . 12 (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ ℕ0)
63 eqid 2621 . . . . . . . . . . . . . 14 (deg‘(ℂ × {𝐴})) = (deg‘(ℂ × {𝐴}))
6426, 63dgrub 23911 . . . . . . . . . . . . 13 (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0 ∧ ((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0) → 𝑘 ≤ (deg‘(ℂ × {𝐴})))
65643expia 1264 . . . . . . . . . . . 12 (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(ℂ × {𝐴}))))
6625, 62, 65syl2an 494 . . . . . . . . . . 11 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(ℂ × {𝐴}))))
67 0dgr 23922 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0)
6867ad3antrrr 765 . . . . . . . . . . . . 13 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (deg‘(ℂ × {𝐴})) = 0)
6968breq2d 4630 . . . . . . . . . . . 12 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ (deg‘(ℂ × {𝐴})) ↔ 𝑘 ≤ 0))
7062adantl 482 . . . . . . . . . . . . 13 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → 𝑘 ∈ ℕ0)
71 nn0le0eq0 11273 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 → (𝑘 ≤ 0 ↔ 𝑘 = 0))
7270, 71syl 17 . . . . . . . . . . . 12 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ 0 ↔ 𝑘 = 0))
7369, 72bitrd 268 . . . . . . . . . . 11 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ (deg‘(ℂ × {𝐴})) ↔ 𝑘 = 0))
7466, 73sylibd 229 . . . . . . . . . 10 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 = 0))
75 id 22 . . . . . . . . . . 11 (𝑘 = 0 → 𝑘 = 0)
76 0z 11340 . . . . . . . . . . . 12 0 ∈ ℤ
77 elfz3 12301 . . . . . . . . . . . 12 (0 ∈ ℤ → 0 ∈ (0...0))
7876, 77ax-mp 5 . . . . . . . . . . 11 0 ∈ (0...0)
7975, 78syl6eqel 2706 . . . . . . . . . 10 (𝑘 = 0 → 𝑘 ∈ (0...0))
8074, 79syl6 35 . . . . . . . . 9 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ∈ (0...0)))
8180necon1bd 2808 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (¬ 𝑘 ∈ (0...0) → ((coeff‘(ℂ × {𝐴}))‘𝑘) = 0))
8259, 81mpd 15 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ((coeff‘(ℂ × {𝐴}))‘𝑘) = 0)
8382oveq1d 6625 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))) = (0 · ((coeff‘𝐹)‘(𝑛𝑘))))
8418adantr 481 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (coeff‘𝐹):ℕ0⟶ℂ)
85 fznn0sub 12323 . . . . . . . . 9 (𝑘 ∈ (0...𝑛) → (𝑛𝑘) ∈ ℕ0)
8660, 85syl 17 . . . . . . . 8 (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → (𝑛𝑘) ∈ ℕ0)
87 ffvelrn 6318 . . . . . . . 8 (((coeff‘𝐹):ℕ0⟶ℂ ∧ (𝑛𝑘) ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛𝑘)) ∈ ℂ)
8884, 86, 87syl2an 494 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ((coeff‘𝐹)‘(𝑛𝑘)) ∈ ℂ)
8988mul02d 10186 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (0 · ((coeff‘𝐹)‘(𝑛𝑘))) = 0)
9083, 89eqtrd 2655 . . . . 5 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))) = 0)
91 fzfid 12720 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...𝑛) ∈ Fin)
9245, 57, 90, 91fsumss 14397 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → Σ𝑘 ∈ (0...0)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))) = Σ𝑘 ∈ (0...𝑛)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))))
9351fsum1 14417 . . . . 5 ((0 ∈ ℤ ∧ (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
9476, 55, 93sylancr 694 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → Σ𝑘 ∈ (0...0)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
9541, 92, 943eqtr2d 2661 . . 3 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹))‘𝑛) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
96 simpl 473 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐴 ∈ ℂ)
97 eqidd 2622 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘𝑛) = ((coeff‘𝐹)‘𝑛))
9822, 96, 20, 97ofc1 6880 . . 3 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((ℕ0 × {𝐴}) ∘𝑓 · (coeff‘𝐹))‘𝑛) = (𝐴 · ((coeff‘𝐹)‘𝑛)))
9938, 95, 983eqtr4d 2665 . 2 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹))‘𝑛) = (((ℕ0 × {𝐴}) ∘𝑓 · (coeff‘𝐹))‘𝑛))
10011, 24, 99eqfnfvd 6275 1 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)) = ((ℕ0 × {𝐴}) ∘𝑓 · (coeff‘𝐹)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  Vcvv 3189   ∖ cdif 3556   ⊆ wss 3559  {csn 4153   class class class wbr 4618   × cxp 5077   Fn wfn 5847  ⟶wf 5848  ‘cfv 5852  (class class class)co 6610   ∘𝑓 cof 6855  ℂcc 9886  0cc0 9888   · cmul 9893   ≤ cle 10027   − cmin 10218  ℕ0cn0 11244  ℤcz 11329  ℤ≥cuz 11639  ...cfz 12276  Σcsu 14358  Polycply 23861  coeffccoe 23863  degcdgr 23864 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 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966  ax-addf 9967 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 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-inf 8301  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-z 11330  df-uz 11640  df-rp 11785  df-fz 12277  df-fzo 12415  df-fl 12541  df-seq 12750  df-exp 12809  df-hash 13066  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-clim 14161  df-rlim 14162  df-sum 14359  df-0p 23360  df-ply 23865  df-coe 23867  df-dgr 23868 This theorem is referenced by:  coe0  23933  coesub  23934  mpaaeu  37236
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