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‘(𝐹 ∘𝑓 − 𝐺)) = (𝐴 ∘𝑓 − 𝐵))

Proof of Theorem coesub

Step	Hyp	Ref	Expression
1		plyssc 24175	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		simpl 474	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
3	1, 2	sseldi 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))
7	4, 5, 6	mp2an 710	. . . . 5 ⊢ (ℂ × {-1}) ∈ (Poly‘ℂ)
8		simpr 479	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
9	1, 8	sseldi 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))
11	7, 9, 10	sylancr 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))
14	12, 13	coeadd 24226	. . . 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 696	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴 ∘𝑓 + (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺))))
16		coemulc 24230	. . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝐺 ∈ (Poly‘ℂ)) → (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = ((ℕ0 × {-1}) ∘𝑓 · (coeff‘𝐺)))
17	5, 9, 16	sylancr 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)
19	18	oveq2i 6825	. . . . 5 ⊢ ((ℕ0 × {-1}) ∘𝑓 · 𝐵) = ((ℕ0 × {-1}) ∘𝑓 · (coeff‘𝐺))
20	17, 19	syl6eqr 2812	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺)) = ((ℕ0 × {-1}) ∘𝑓 · 𝐵))
21	20	oveq2d 6830	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 ∘𝑓 + (coeff‘((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴 ∘𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)))
22	15, 21	eqtrd 2794	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (𝐴 ∘𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)))
23		cnex 10229	. . . . 5 ⊢ ℂ ∈ V
24	23	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V)
25		plyf 24173	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
26	25	adantr 472	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
27		plyf 24173	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
28	27	adantl 473	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ)
29		ofnegsub 11230	. . . 4 ⊢ ((ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ ∧ 𝐺:ℂ⟶ℂ) → (𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺))
30	24, 26, 28, 29	syl3anc 1477	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺))
31	30	fveq2d 6357	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺))) = (coeff‘(𝐹 ∘𝑓 − 𝐺)))
32		nn0ex 11510	. . . 4 ⊢ ℕ0 ∈ V
33	32	a1i 11	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
34	12	coef3 24207	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
35	34	adantr 472	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
36	18	coef3 24207	. . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
37	36	adantl 473	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
38		ofnegsub 11230	. . 3 ⊢ ((ℕ0 ∈ V ∧ 𝐴:ℕ0⟶ℂ ∧ 𝐵:ℕ0⟶ℂ) → (𝐴 ∘𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)) = (𝐴 ∘𝑓 − 𝐵))
39	33, 35, 37, 38	syl3anc 1477	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 ∘𝑓 + ((ℕ0 × {-1}) ∘𝑓 · 𝐵)) = (𝐴 ∘𝑓 − 𝐵))
40	22, 31, 39	3eqtr3d 2802	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘𝑓 − 𝐺)) = (𝐴 ∘𝑓 − 𝐵))

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  ℂcc 10146  1c1 10149   + caddc 10151   · cmul 10153   − cmin 10478  -cneg 10479  ℕ₀cn0 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