Theorem plymul0or 26323

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 26273	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
2		dgrcl 26273	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
3		nn0addcl 12563	. . . . . . 7 ⊢ (((deg‘𝐹) ∈ ℕ0 ∧ (deg‘𝐺) ∈ ℕ0) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
4	1, 2, 3	syl2an 596	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
5		c0ex 11256	. . . . . . 7 ⊢ 0 ∈ V
6	5	fvconst2 7225	. . . . . 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 6905	. . . . . . . 8 ⊢ ((𝐹 ∘f · 𝐺) = 0𝑝 → (coeff‘(𝐹 ∘f · 𝐺)) = (coeff‘0𝑝))
9		coe0 26296	. . . . . . . 8 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
10	8, 9	eqtrdi 2792	. . . . . . 7 ⊢ ((𝐹 ∘f · 𝐺) = 0𝑝 → (coeff‘(𝐹 ∘f · 𝐺)) = (ℕ0 × {0}))
11	10	fveq1d 6907	. . . . . 6 ⊢ ((𝐹 ∘f · 𝐺) = 0𝑝 → ((coeff‘(𝐹 ∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))))
12	11	eqeq1d 2738	. . . . 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 247	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 → ((coeff‘(𝐹 ∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0))
14		eqid 2736	. . . . . . 7 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
15		eqid 2736	. . . . . . 7 ⊢ (coeff‘𝐺) = (coeff‘𝐺)
16		eqid 2736	. . . . . . 7 ⊢ (deg‘𝐹) = (deg‘𝐹)
17		eqid 2736	. . . . . . 7 ⊢ (deg‘𝐺) = (deg‘𝐺)
18	14, 15, 16, 17	coemulhi 26294	. . . . . 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 2738	. . . . 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 26272	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
21	20	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
22	1	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℕ0)
23	21, 22	ffvelcdmd 7104	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐹)‘(deg‘𝐹)) ∈ ℂ)
24	15	coef3 26272	. . . . . . . 8 ⊢ (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ)
25	24	adantl 481	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐺):ℕ0⟶ℂ)
26	2	adantl 481	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐺) ∈ ℕ0)
27	25, 26	ffvelcdmd 7104	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐺)‘(deg‘𝐺)) ∈ ℂ)
28	23, 27	mul0ord 11914	. . . . 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 279	. . . 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 239	. . 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 26306	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
32	31	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
33	17, 15	dgreq0 26306	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
34	33	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
35	32, 34	orbi12d 918	. . 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 259	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 → (𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝)))
37		cnex 11237	. . . . . . 7 ⊢ ℂ ∈ V
38	37	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V)
39		plyf 26238	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
40	39	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ)
41		0cnd 11255	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ∈ ℂ)
42		mul02 11440	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
43	42	adantl 481	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0)
44	38, 40, 41, 41, 43	caofid2 7734	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℂ × {0}) ∘f · 𝐺) = (ℂ × {0}))
45		id 22	. . . . . . . 8 ⊢ (𝐹 = 0𝑝 → 𝐹 = 0𝑝)
46		df-0p 25706	. . . . . . . 8 ⊢ 0𝑝 = (ℂ × {0})
47	45, 46	eqtrdi 2792	. . . . . . 7 ⊢ (𝐹 = 0𝑝 → 𝐹 = (ℂ × {0}))
48	47	oveq1d 7447	. . . . . 6 ⊢ (𝐹 = 0𝑝 → (𝐹 ∘f · 𝐺) = ((ℂ × {0}) ∘f · 𝐺))
49	48	eqeq1d 2738	. . . . 5 ⊢ (𝐹 = 0𝑝 → ((𝐹 ∘f · 𝐺) = (ℂ × {0}) ↔ ((ℂ × {0}) ∘f · 𝐺) = (ℂ × {0})))
50	44, 49	syl5ibrcom 247	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 → (𝐹 ∘f · 𝐺) = (ℂ × {0})))
51		plyf 26238	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
52	51	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
53		mul01 11441	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
54	53	adantl 481	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
55	38, 52, 41, 41, 54	caofid1 7733	. . . . 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 2792	. . . . . . 7 ⊢ (𝐺 = 0𝑝 → 𝐺 = (ℂ × {0}))
58	57	oveq2d 7448	. . . . . 6 ⊢ (𝐺 = 0𝑝 → (𝐹 ∘f · 𝐺) = (𝐹 ∘f · (ℂ × {0})))
59	58	eqeq1d 2738	. . . . 5 ⊢ (𝐺 = 0𝑝 → ((𝐹 ∘f · 𝐺) = (ℂ × {0}) ↔ (𝐹 ∘f · (ℂ × {0})) = (ℂ × {0})))
60	55, 59	syl5ibrcom 247	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 → (𝐹 ∘f · 𝐺) = (ℂ × {0})))
61	50, 60	jaod 859	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝) → (𝐹 ∘f · 𝐺) = (ℂ × {0})))
62	46	eqeq2i 2749	. . 3 ⊢ ((𝐹 ∘f · 𝐺) = 0𝑝 ↔ (𝐹 ∘f · 𝐺) = (ℂ × {0}))
63	61, 62	imbitrrdi 252	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝) → (𝐹 ∘f · 𝐺) = 0𝑝))
64	36, 63	impbid 212	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107  Vcvv 3479  {csn 4625   × cxp 5682  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ∘_f cof 7696  ℂcc 11154  0cc0 11156   + caddc 11159   · cmul 11161  ℕ₀cn0 12528  0_𝑝c0p 25705  Polycply 26224  coeffccoe 26226  degcdgr 26227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-rp 13036  df-fz 13549  df-fzo 13696  df-fl 13833  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-rlim 15526  df-sum 15724  df-0p 25706  df-ply 26228  df-coe 26230  df-dgr 26231

This theorem is referenced by:  plydiveu  26341  quotcan  26352  vieta1lem1  26353  vieta1lem2  26354