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Theorem coesub 24058
 Description: The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
coesub.1 𝐴 = (coeff‘𝐹)
coesub.2 𝐵 = (coeff‘𝐺)
Assertion
Ref Expression
coesub ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓𝐺)) = (𝐴𝑓𝐵))

Proof of Theorem coesub
StepHypRef Expression
1 plyssc 24001 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simpl 472 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
31, 2sseldi 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))
74, 5, 6mp2an 708 . . . . 5 (ℂ × {-1}) ∈ (Poly‘ℂ)
8 simpr 476 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
91, 8sseldi 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))
117, 9, 10sylancr 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))
1412, 13coeadd 24052 . . . 4 ((𝐹 ∈ (Poly‘ℂ) ∧ ((ℂ × {-1}) ∘𝑓 · 𝐺) ∈ (Poly‘ℂ)) → (coeff‘(𝐹𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴𝑓 + (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺))))
153, 11, 14syl2anc 694 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴𝑓 + (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺))))
16 coemulc 24056 . . . . . 6 ((-1 ∈ ℂ ∧ 𝐺 ∈ (Poly‘ℂ)) → (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = ((ℕ0 × {-1}) ∘𝑓 · (coeff‘𝐺)))
175, 9, 16sylancr 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)
1918oveq2i 6701 . . . . 5 ((ℕ0 × {-1}) ∘𝑓 · 𝐵) = ((ℕ0 × {-1}) ∘𝑓 · (coeff‘𝐺))
2017, 19syl6eqr 2703 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = ((ℕ0 × {-1}) ∘𝑓 · 𝐵))
2120oveq2d 6706 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴𝑓 + (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)))
2215, 21eqtrd 2685 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)))
23 cnex 10055 . . . . 5 ℂ ∈ V
2423a1i 11 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V)
25 plyf 23999 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
2625adantr 480 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
27 plyf 23999 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
2827adantl 481 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ)
29 ofnegsub 11056 . . . 4 ((ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ ∧ 𝐺:ℂ⟶ℂ) → (𝐹𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹𝑓𝐺))
3024, 26, 28, 29syl3anc 1366 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹𝑓𝐺))
3130fveq2d 6233 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (coeff‘(𝐹𝑓𝐺)))
32 nn0ex 11336 . . . 4 0 ∈ V
3332a1i 11 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
3412coef3 24033 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
3534adantr 480 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
3618coef3 24033 . . . 4 (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
3736adantl 481 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
38 ofnegsub 11056 . . 3 ((ℕ0 ∈ V ∧ 𝐴:ℕ0⟶ℂ ∧ 𝐵:ℕ0⟶ℂ) → (𝐴𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)) = (𝐴𝑓𝐵))
3933, 35, 37, 38syl3anc 1366 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)) = (𝐴𝑓𝐵))
4022, 31, 393eqtr3d 2693 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓𝐺)) = (𝐴𝑓𝐵))
 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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