Theorem plymul0or 24877

Description: Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)

Assertion

Ref	Expression
plymul0or	⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝)))

Proof of Theorem plymul0or

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dgrcl 24830	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
2		dgrcl 24830	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
3		nn0addcl 11920	. . . . . . 7 ⊢ (((deg‘𝐹) ∈ ℕ0 ∧ (deg‘𝐺) ∈ ℕ0) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
4	1, 2, 3	syl2an 598	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
5		c0ex 10624	. . . . . . 7 ⊢ 0 ∈ V
6	5	fvconst2 6943	. . . . . 6 ⊢ (((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0 → ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))) = 0)
7	4, 6	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))) = 0)
8		fveq2 6645	. . . . . . . 8 ⊢ ((𝐹 ∘f · 𝐺) = 0𝑝 → (coeff‘(𝐹 ∘f · 𝐺)) = (coeff‘0𝑝))
9		coe0 24853	. . . . . . . 8 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
10	8, 9	eqtrdi 2849	. . . . . . 7 ⊢ ((𝐹 ∘f · 𝐺) = 0𝑝 → (coeff‘(𝐹 ∘f · 𝐺)) = (ℕ0 × {0}))
11	10	fveq1d 6647	. . . . . 6 ⊢ ((𝐹 ∘f · 𝐺) = 0𝑝 → ((coeff‘(𝐹 ∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))))
12	11	eqeq1d 2800	. . . . 5 ⊢ ((𝐹 ∘f · 𝐺) = 0𝑝 → (((coeff‘(𝐹 ∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0 ↔ ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))) = 0))
13	7, 12	syl5ibrcom 250	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 → ((coeff‘(𝐹 ∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0))
14		eqid 2798	. . . . . . 7 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
15		eqid 2798	. . . . . . 7 ⊢ (coeff‘𝐺) = (coeff‘𝐺)
16		eqid 2798	. . . . . . 7 ⊢ (deg‘𝐹) = (deg‘𝐹)
17		eqid 2798	. . . . . . 7 ⊢ (deg‘𝐺) = (deg‘𝐺)
18	14, 15, 16, 17	coemulhi 24851	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹 ∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = (((coeff‘𝐹)‘(deg‘𝐹)) · ((coeff‘𝐺)‘(deg‘𝐺))))
19	18	eqeq1d 2800	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (((coeff‘(𝐹 ∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0 ↔ (((coeff‘𝐹)‘(deg‘𝐹)) · ((coeff‘𝐺)‘(deg‘𝐺))) = 0))
20	14	coef3 24829	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
21	20	adantr 484	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
22	1	adantr 484	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℕ0)
23	21, 22	ffvelrnd 6829	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐹)‘(deg‘𝐹)) ∈ ℂ)
24	15	coef3 24829	. . . . . . . 8 ⊢ (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ)
25	24	adantl 485	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐺):ℕ0⟶ℂ)
26	2	adantl 485	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐺) ∈ ℕ0)
27	25, 26	ffvelrnd 6829	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐺)‘(deg‘𝐺)) ∈ ℂ)
28	23, 27	mul0ord 11279	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((((coeff‘𝐹)‘(deg‘𝐹)) · ((coeff‘𝐺)‘(deg‘𝐺))) = 0 ↔ (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
29	19, 28	bitrd 282	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (((coeff‘(𝐹 ∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0 ↔ (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
30	13, 29	sylibd 242	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 → (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
31	16, 14	dgreq0 24862	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
32	31	adantr 484	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
33	17, 15	dgreq0 24862	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
34	33	adantl 485	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
35	32, 34	orbi12d 916	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝) ↔ (((coeff‘𝐹)‘(deg‘𝐹)) = 0 ∨ ((coeff‘𝐺)‘(deg‘𝐺)) = 0)))
36	30, 35	sylibrd 262	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 → (𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝)))
37		cnex 10607	. . . . . . 7 ⊢ ℂ ∈ V
38	37	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V)
39		plyf 24795	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
40	39	adantl 485	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ)
41		0cnd 10623	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ∈ ℂ)
42		mul02 10807	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
43	42	adantl 485	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0)
44	38, 40, 41, 41, 43	caofid2 7420	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℂ × {0}) ∘f · 𝐺) = (ℂ × {0}))
45		id 22	. . . . . . . 8 ⊢ (𝐹 = 0𝑝 → 𝐹 = 0𝑝)
46		df-0p 24274	. . . . . . . 8 ⊢ 0𝑝 = (ℂ × {0})
47	45, 46	eqtrdi 2849	. . . . . . 7 ⊢ (𝐹 = 0𝑝 → 𝐹 = (ℂ × {0}))
48	47	oveq1d 7150	. . . . . 6 ⊢ (𝐹 = 0𝑝 → (𝐹 ∘f · 𝐺) = ((ℂ × {0}) ∘f · 𝐺))
49	48	eqeq1d 2800	. . . . 5 ⊢ (𝐹 = 0𝑝 → ((𝐹 ∘f · 𝐺) = (ℂ × {0}) ↔ ((ℂ × {0}) ∘f · 𝐺) = (ℂ × {0})))
50	44, 49	syl5ibrcom 250	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 → (𝐹 ∘f · 𝐺) = (ℂ × {0})))
51		plyf 24795	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
52	51	adantr 484	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
53		mul01 10808	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
54	53	adantl 485	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
55	38, 52, 41, 41, 54	caofid1 7419	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f · (ℂ × {0})) = (ℂ × {0}))
56		id 22	. . . . . . . 8 ⊢ (𝐺 = 0𝑝 → 𝐺 = 0𝑝)
57	56, 46	eqtrdi 2849	. . . . . . 7 ⊢ (𝐺 = 0𝑝 → 𝐺 = (ℂ × {0}))
58	57	oveq2d 7151	. . . . . 6 ⊢ (𝐺 = 0𝑝 → (𝐹 ∘f · 𝐺) = (𝐹 ∘f · (ℂ × {0})))
59	58	eqeq1d 2800	. . . . 5 ⊢ (𝐺 = 0𝑝 → ((𝐹 ∘f · 𝐺) = (ℂ × {0}) ↔ (𝐹 ∘f · (ℂ × {0})) = (ℂ × {0})))
60	55, 59	syl5ibrcom 250	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 → (𝐹 ∘f · 𝐺) = (ℂ × {0})))
61	50, 60	jaod 856	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝) → (𝐹 ∘f · 𝐺) = (ℂ × {0})))
62	46	eqeq2i 2811	. . 3 ⊢ ((𝐹 ∘f · 𝐺) = 0𝑝 ↔ (𝐹 ∘f · 𝐺) = (ℂ × {0}))
63	61, 62	syl6ibr 255	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝) → (𝐹 ∘f · 𝐺) = 0𝑝))
64	36, 63	impbid 215	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3441  {csn 4525   × cxp 5517  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∘_f cof 7387  ℂcc 10524  0cc0 10526   + caddc 10529   · cmul 10531  ℕ₀cn0 11885  0_𝑝c0p 24273  Polycply 24781  coeffccoe 24783  degcdgr 24784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-inf 8891  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-rlim 14838  df-sum 15035  df-0p 24274  df-ply 24785  df-coe 24787  df-dgr 24788

This theorem is referenced by:  plydiveu  24894  quotcan  24905  vieta1lem1  24906  vieta1lem2  24907