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Mirrors > Home > MPE Home > Th. List > coesub | Structured version Visualization version GIF version |
Description: The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
coesub.1 | ⊢ 𝐴 = (coeff‘𝐹) |
coesub.2 | ⊢ 𝐵 = (coeff‘𝐺) |
Ref | Expression |
---|---|
coesub | ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘𝑓 − 𝐺)) = (𝐴 ∘𝑓 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plyssc 24001 | . . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
2 | simpl 472 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆)) | |
3 | 1, 2 | sseldi 3634 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘ℂ)) |
4 | ssid 3657 | . . . . . 6 ⊢ ℂ ⊆ ℂ | |
5 | neg1cn 11162 | . . . . . 6 ⊢ -1 ∈ ℂ | |
6 | plyconst 24007 | . . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ -1 ∈ ℂ) → (ℂ × {-1}) ∈ (Poly‘ℂ)) | |
7 | 4, 5, 6 | mp2an 708 | . . . . 5 ⊢ (ℂ × {-1}) ∈ (Poly‘ℂ) |
8 | simpr 476 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆)) | |
9 | 1, 8 | sseldi 3634 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘ℂ)) |
10 | plymulcl 24022 | . . . . 5 ⊢ (((ℂ × {-1}) ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → ((ℂ × {-1}) ∘𝑓 · 𝐺) ∈ (Poly‘ℂ)) | |
11 | 7, 9, 10 | sylancr 696 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℂ × {-1}) ∘𝑓 · 𝐺) ∈ (Poly‘ℂ)) |
12 | coesub.1 | . . . . 5 ⊢ 𝐴 = (coeff‘𝐹) | |
13 | eqid 2651 | . . . . 5 ⊢ (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)) | |
14 | 12, 13 | coeadd 24052 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ ((ℂ × {-1}) ∘𝑓 · 𝐺) ∈ (Poly‘ℂ)) → (coeff‘(𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴 ∘𝑓 + (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)))) |
15 | 3, 11, 14 | syl2anc 694 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴 ∘𝑓 + (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)))) |
16 | coemulc 24056 | . . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝐺 ∈ (Poly‘ℂ)) → (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = ((ℕ0 × {-1}) ∘𝑓 · (coeff‘𝐺))) | |
17 | 5, 9, 16 | sylancr 696 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = ((ℕ0 × {-1}) ∘𝑓 · (coeff‘𝐺))) |
18 | coesub.2 | . . . . . 6 ⊢ 𝐵 = (coeff‘𝐺) | |
19 | 18 | oveq2i 6701 | . . . . 5 ⊢ ((ℕ0 × {-1}) ∘𝑓 · 𝐵) = ((ℕ0 × {-1}) ∘𝑓 · (coeff‘𝐺)) |
20 | 17, 19 | syl6eqr 2703 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = ((ℕ0 × {-1}) ∘𝑓 · 𝐵)) |
21 | 20 | oveq2d 6706 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 ∘𝑓 + (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴 ∘𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵))) |
22 | 15, 21 | eqtrd 2685 | . 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴 ∘𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵))) |
23 | cnex 10055 | . . . . 5 ⊢ ℂ ∈ V | |
24 | 23 | a1i 11 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V) |
25 | plyf 23999 | . . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
26 | 25 | adantr 480 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ) |
27 | plyf 23999 | . . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ) | |
28 | 27 | adantl 481 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ) |
29 | ofnegsub 11056 | . . . 4 ⊢ ((ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ ∧ 𝐺:ℂ⟶ℂ) → (𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺)) | |
30 | 24, 26, 28, 29 | syl3anc 1366 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺)) |
31 | 30 | fveq2d 6233 | . 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (coeff‘(𝐹 ∘𝑓 − 𝐺))) |
32 | nn0ex 11336 | . . . 4 ⊢ ℕ0 ∈ V | |
33 | 32 | a1i 11 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V) |
34 | 12 | coef3 24033 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
35 | 34 | adantr 480 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ) |
36 | 18 | coef3 24033 | . . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ) |
37 | 36 | adantl 481 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ) |
38 | ofnegsub 11056 | . . 3 ⊢ ((ℕ0 ∈ V ∧ 𝐴:ℕ0⟶ℂ ∧ 𝐵:ℕ0⟶ℂ) → (𝐴 ∘𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)) = (𝐴 ∘𝑓 − 𝐵)) | |
39 | 33, 35, 37, 38 | syl3anc 1366 | . 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 ∘𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)) = (𝐴 ∘𝑓 − 𝐵)) |
40 | 22, 31, 39 | 3eqtr3d 2693 | 1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘𝑓 − 𝐺)) = (𝐴 ∘𝑓 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ⊆ wss 3607 {csn 4210 × cxp 5141 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 ∘𝑓 cof 6937 ℂcc 9972 1c1 9975 + caddc 9977 · cmul 9979 − cmin 10304 -cneg 10305 ℕ0cn0 11330 Polycply 23985 coeffccoe 23987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-rlim 14264 df-sum 14461 df-0p 23482 df-ply 23989 df-coe 23991 df-dgr 23992 |
This theorem is referenced by: dgrcolem2 24075 plydivlem4 24096 plydiveu 24098 vieta1lem2 24111 dgrsub2 38022 |
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