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Theorem coesub 24232
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 24175 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simpl 474 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
31, 2sseldi 3742 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘ℂ))
4 ssid 3765 . . . . . 6 ℂ ⊆ ℂ
5 neg1cn 11336 . . . . . 6 -1 ∈ ℂ
6 plyconst 24181 . . . . . 6 ((ℂ ⊆ ℂ ∧ -1 ∈ ℂ) → (ℂ × {-1}) ∈ (Poly‘ℂ))
74, 5, 6mp2an 710 . . . . 5 (ℂ × {-1}) ∈ (Poly‘ℂ)
8 simpr 479 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
91, 8sseldi 3742 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘ℂ))
10 plymulcl 24196 . . . . 5 (((ℂ × {-1}) ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ)) → ((ℂ × {-1}) ∘𝑓 · 𝐺) ∈ (Poly‘ℂ))
117, 9, 10sylancr 698 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℂ × {-1}) ∘𝑓 · 𝐺) ∈ (Poly‘ℂ))
12 coesub.1 . . . . 5 𝐴 = (coeff‘𝐹)
13 eqid 2760 . . . . 5 (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺))
1412, 13coeadd 24226 . . . 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 696 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴𝑓 + (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺))))
16 coemulc 24230 . . . . . 6 ((-1 ∈ ℂ ∧ 𝐺 ∈ (Poly‘ℂ)) → (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = ((ℕ0 × {-1}) ∘𝑓 · (coeff‘𝐺)))
175, 9, 16sylancr 698 . . . . 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 6825 . . . . 5 ((ℕ0 × {-1}) ∘𝑓 · 𝐵) = ((ℕ0 × {-1}) ∘𝑓 · (coeff‘𝐺))
2017, 19syl6eqr 2812 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = ((ℕ0 × {-1}) ∘𝑓 · 𝐵))
2120oveq2d 6830 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴𝑓 + (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)))
2215, 21eqtrd 2794 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)))
23 cnex 10229 . . . . 5 ℂ ∈ V
2423a1i 11 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V)
25 plyf 24173 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
2625adantr 472 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
27 plyf 24173 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
2827adantl 473 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ)
29 ofnegsub 11230 . . . 4 ((ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ ∧ 𝐺:ℂ⟶ℂ) → (𝐹𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹𝑓𝐺))
3024, 26, 28, 29syl3anc 1477 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹𝑓𝐺))
3130fveq2d 6357 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (coeff‘(𝐹𝑓𝐺)))
32 nn0ex 11510 . . . 4 0 ∈ V
3332a1i 11 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
3412coef3 24207 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
3534adantr 472 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
3618coef3 24207 . . . 4 (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
3736adantl 473 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
38 ofnegsub 11230 . . 3 ((ℕ0 ∈ V ∧ 𝐴:ℕ0⟶ℂ ∧ 𝐵:ℕ0⟶ℂ) → (𝐴𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)) = (𝐴𝑓𝐵))
3933, 35, 37, 38syl3anc 1477 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)) = (𝐴𝑓𝐵))
4022, 31, 393eqtr3d 2802 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹𝑓𝐺)) = (𝐴𝑓𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  wss 3715  {csn 4321   × cxp 5264  wf 6045  cfv 6049  (class class class)co 6814  𝑓 cof 7061  cc 10146  1c1 10149   + caddc 10151   · cmul 10153  cmin 10478  -cneg 10479  0cn0 11504  Polycply 24159  coeffccoe 24161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226  ax-addf 10227
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-inf 8516  df-oi 8582  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-z 11590  df-uz 11900  df-rp 12046  df-fz 12540  df-fzo 12680  df-fl 12807  df-seq 13016  df-exp 13075  df-hash 13332  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438  df-rlim 14439  df-sum 14636  df-0p 23656  df-ply 24163  df-coe 24165  df-dgr 24166
This theorem is referenced by:  dgrcolem2  24249  plydivlem4  24270  plydiveu  24272  vieta1lem2  24285  dgrsub2  38225
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