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Theorem coemulc 24302
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 3783 . . . . 5 ℂ ⊆ ℂ
2 plyconst 24253 . . . . 5 ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
31, 2mpan 681 . . . 4 (𝐴 ∈ ℂ → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
4 plyssc 24247 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
54sseli 3757 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
6 plymulcl 24268 . . . 4 (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝐹 ∈ (Poly‘ℂ)) → ((ℂ × {𝐴}) ∘𝑓 · 𝐹) ∈ (Poly‘ℂ))
73, 5, 6syl2an 589 . . 3 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ × {𝐴}) ∘𝑓 · 𝐹) ∈ (Poly‘ℂ))
8 eqid 2765 . . . 4 (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)) = (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹))
98coef3 24279 . . 3 (((ℂ × {𝐴}) ∘𝑓 · 𝐹) ∈ (Poly‘ℂ) → (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)):ℕ0⟶ℂ)
10 ffn 6223 . . 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 6274 . . . . 5 (𝐴 ∈ ℂ → (ℕ0 × {𝐴}):ℕ0⟶{𝐴})
1312adantr 472 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ0 × {𝐴}):ℕ0⟶{𝐴})
1413ffnd 6224 . . 3 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ0 × {𝐴}) Fn ℕ0)
15 eqid 2765 . . . . . 6 (coeff‘𝐹) = (coeff‘𝐹)
1615coef3 24279 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
1716adantl 473 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
1817ffnd 6224 . . 3 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹) Fn ℕ0)
19 nn0ex 11545 . . . 4 0 ∈ V
2019a1i 11 . . 3 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
21 inidm 3982 . . 3 (ℕ0 ∩ ℕ0) = ℕ0
2214, 18, 20, 20, 21offn 7106 . 2 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℕ0 × {𝐴}) ∘𝑓 · (coeff‘𝐹)) Fn ℕ0)
233ad2antrr 717 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
24 eqid 2765 . . . . . . 7 (coeff‘(ℂ × {𝐴})) = (coeff‘(ℂ × {𝐴}))
2524coefv0 24295 . . . . . 6 ((ℂ × {𝐴}) ∈ (Poly‘ℂ) → ((ℂ × {𝐴})‘0) = ((coeff‘(ℂ × {𝐴}))‘0))
2623, 25syl 17 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ × {𝐴})‘0) = ((coeff‘(ℂ × {𝐴}))‘0))
27 simpll 783 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ ℂ)
28 0cn 10285 . . . . . 6 0 ∈ ℂ
29 fvconst2g 6660 . . . . . 6 ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ((ℂ × {𝐴})‘0) = 𝐴)
3027, 28, 29sylancl 580 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ × {𝐴})‘0) = 𝐴)
3126, 30eqtr3d 2801 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(ℂ × {𝐴}))‘0) = 𝐴)
32 simpr 477 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
3332nn0cnd 11600 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℂ)
3433subid1d 10635 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (𝑛 − 0) = 𝑛)
3534fveq2d 6379 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 0)) = ((coeff‘𝐹)‘𝑛))
3631, 35oveq12d 6860 . . 3 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) = (𝐴 · ((coeff‘𝐹)‘𝑛)))
375ad2antlr 718 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℂ))
3824, 15coemul 24299 . . . . 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))))
3923, 37, 32, 38syl3anc 1490 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹))‘𝑛) = Σ𝑘 ∈ (0...𝑛)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))))
40 nn0uz 11922 . . . . . . 7 0 = (ℤ‘0)
4132, 40syl6eleq 2854 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ (ℤ‘0))
42 fzss2 12588 . . . . . 6 (𝑛 ∈ (ℤ‘0) → (0...0) ⊆ (0...𝑛))
4341, 42syl 17 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...0) ⊆ (0...𝑛))
44 elfz1eq 12559 . . . . . . . 8 (𝑘 ∈ (0...0) → 𝑘 = 0)
4544adantl 473 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → 𝑘 = 0)
46 fveq2 6375 . . . . . . . 8 (𝑘 = 0 → ((coeff‘(ℂ × {𝐴}))‘𝑘) = ((coeff‘(ℂ × {𝐴}))‘0))
47 oveq2 6850 . . . . . . . . 9 (𝑘 = 0 → (𝑛𝑘) = (𝑛 − 0))
4847fveq2d 6379 . . . . . . . 8 (𝑘 = 0 → ((coeff‘𝐹)‘(𝑛𝑘)) = ((coeff‘𝐹)‘(𝑛 − 0)))
4946, 48oveq12d 6860 . . . . . . 7 (𝑘 = 0 → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
5045, 49syl 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))))
5117ffvelrnda 6549 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘𝑛) ∈ ℂ)
5227, 51mulcld 10314 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (𝐴 · ((coeff‘𝐹)‘𝑛)) ∈ ℂ)
5336, 52eqeltrd 2844 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ)
5453adantr 472 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ)
5550, 54eqeltrd 2844 . . . . 5 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))) ∈ ℂ)
56 eldifn 3895 . . . . . . . . 9 (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → ¬ 𝑘 ∈ (0...0))
5756adantl 473 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ¬ 𝑘 ∈ (0...0))
58 eldifi 3894 . . . . . . . . . . . . 13 (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ (0...𝑛))
59 elfznn0 12640 . . . . . . . . . . . . 13 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
6058, 59syl 17 . . . . . . . . . . . 12 (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ ℕ0)
61 eqid 2765 . . . . . . . . . . . . . 14 (deg‘(ℂ × {𝐴})) = (deg‘(ℂ × {𝐴}))
6224, 61dgrub 24281 . . . . . . . . . . . . 13 (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0 ∧ ((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0) → 𝑘 ≤ (deg‘(ℂ × {𝐴})))
63623expia 1150 . . . . . . . . . . . 12 (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(ℂ × {𝐴}))))
6423, 60, 63syl2an 589 . . . . . . . . . . 11 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(ℂ × {𝐴}))))
65 0dgr 24292 . . . . . . . . . . . . . 14 (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0)
6665ad3antrrr 721 . . . . . . . . . . . . 13 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (deg‘(ℂ × {𝐴})) = 0)
6766breq2d 4821 . . . . . . . . . . . 12 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ (deg‘(ℂ × {𝐴})) ↔ 𝑘 ≤ 0))
6860adantl 473 . . . . . . . . . . . . 13 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → 𝑘 ∈ ℕ0)
69 nn0le0eq0 11568 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ0 → (𝑘 ≤ 0 ↔ 𝑘 = 0))
7068, 69syl 17 . . . . . . . . . . . 12 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ 0 ↔ 𝑘 = 0))
7167, 70bitrd 270 . . . . . . . . . . 11 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ (deg‘(ℂ × {𝐴})) ↔ 𝑘 = 0))
7264, 71sylibd 230 . . . . . . . . . 10 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 = 0))
73 id 22 . . . . . . . . . . 11 (𝑘 = 0 → 𝑘 = 0)
74 0z 11635 . . . . . . . . . . . 12 0 ∈ ℤ
75 elfz3 12558 . . . . . . . . . . . 12 (0 ∈ ℤ → 0 ∈ (0...0))
7674, 75ax-mp 5 . . . . . . . . . . 11 0 ∈ (0...0)
7773, 76syl6eqel 2852 . . . . . . . . . 10 (𝑘 = 0 → 𝑘 ∈ (0...0))
7872, 77syl6 35 . . . . . . . . 9 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ∈ (0...0)))
7978necon1bd 2955 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (¬ 𝑘 ∈ (0...0) → ((coeff‘(ℂ × {𝐴}))‘𝑘) = 0))
8057, 79mpd 15 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ((coeff‘(ℂ × {𝐴}))‘𝑘) = 0)
8180oveq1d 6857 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))) = (0 · ((coeff‘𝐹)‘(𝑛𝑘))))
8217adantr 472 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (coeff‘𝐹):ℕ0⟶ℂ)
83 fznn0sub 12580 . . . . . . . . 9 (𝑘 ∈ (0...𝑛) → (𝑛𝑘) ∈ ℕ0)
8458, 83syl 17 . . . . . . . 8 (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → (𝑛𝑘) ∈ ℕ0)
85 ffvelrn 6547 . . . . . . . 8 (((coeff‘𝐹):ℕ0⟶ℂ ∧ (𝑛𝑘) ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛𝑘)) ∈ ℂ)
8682, 84, 85syl2an 589 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ((coeff‘𝐹)‘(𝑛𝑘)) ∈ ℂ)
8786mul02d 10488 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (0 · ((coeff‘𝐹)‘(𝑛𝑘))) = 0)
8881, 87eqtrd 2799 . . . . 5 ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛𝑘))) = 0)
89 fzfid 12980 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...𝑛) ∈ Fin)
9043, 55, 88, 89fsumss 14743 . . . 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))))
9149fsum1 14763 . . . . 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))))
9274, 53, 91sylancr 581 . . . 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))))
9339, 90, 923eqtr2d 2805 . . 3 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹))‘𝑛) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
94 simpl 474 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐴 ∈ ℂ)
95 eqidd 2766 . . . 4 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘𝑛) = ((coeff‘𝐹)‘𝑛))
9620, 94, 18, 95ofc1 7118 . . 3 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((ℕ0 × {𝐴}) ∘𝑓 · (coeff‘𝐹))‘𝑛) = (𝐴 · ((coeff‘𝐹)‘𝑛)))
9736, 93, 963eqtr4d 2809 . 2 (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹))‘𝑛) = (((ℕ0 × {𝐴}) ∘𝑓 · (coeff‘𝐹))‘𝑛))
9811, 22, 97eqfnfvd 6504 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 197  wa 384   = wceq 1652  wcel 2155  wne 2937  Vcvv 3350  cdif 3729  wss 3732  {csn 4334   class class class wbr 4809   × cxp 5275   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  𝑓 cof 7093  cc 10187  0cc0 10189   · cmul 10194  cle 10329  cmin 10520  0cn0 11538  cz 11624  cuz 11886  ...cfz 12533  Σcsu 14703  Polycply 24231  coeffccoe 24233  degcdgr 24234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-clim 14506  df-rlim 14507  df-sum 14704  df-0p 23728  df-ply 24235  df-coe 24237  df-dgr 24238
This theorem is referenced by:  coe0  24303  coesub  24304  mpaaeu  38329
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