Theorem plymul0or 26195

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 26145	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
2		dgrcl 26145	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
3		nn0addcl 12484	. . . . . . 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 11175	. . . . . . 7 ⊢ 0 ∈ V
6	5	fvconst2 7181	. . . . . 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 6861	. . . . . . . 8 ⊢ ((𝐹 ∘f · 𝐺) = 0𝑝 → (coeff‘(𝐹 ∘f · 𝐺)) = (coeff‘0𝑝))
9		coe0 26168	. . . . . . . 8 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
10	8, 9	eqtrdi 2781	. . . . . . 7 ⊢ ((𝐹 ∘f · 𝐺) = 0𝑝 → (coeff‘(𝐹 ∘f · 𝐺)) = (ℕ0 × {0}))
11	10	fveq1d 6863	. . . . . 6 ⊢ ((𝐹 ∘f · 𝐺) = 0𝑝 → ((coeff‘(𝐹 ∘f · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))))
12	11	eqeq1d 2732	. . . . 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 2730	. . . . . . 7 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
15		eqid 2730	. . . . . . 7 ⊢ (coeff‘𝐺) = (coeff‘𝐺)
16		eqid 2730	. . . . . . 7 ⊢ (deg‘𝐹) = (deg‘𝐹)
17		eqid 2730	. . . . . . 7 ⊢ (deg‘𝐺) = (deg‘𝐺)
18	14, 15, 16, 17	coemulhi 26166	. . . . . 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 2732	. . . . 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 26144	. . . . . . . 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 7060	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐹)‘(deg‘𝐹)) ∈ ℂ)
24	15	coef3 26144	. . . . . . . 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 7060	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐺)‘(deg‘𝐺)) ∈ ℂ)
28	23, 27	mul0ord 11833	. . . . 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 26178	. . . . 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 26178	. . . . 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 11156	. . . . . . 7 ⊢ ℂ ∈ V
38	37	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V)
39		plyf 26110	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
40	39	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ)
41		0cnd 11174	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ∈ ℂ)
42		mul02 11359	. . . . . . 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 7692	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℂ × {0}) ∘f · 𝐺) = (ℂ × {0}))
45		id 22	. . . . . . . 8 ⊢ (𝐹 = 0𝑝 → 𝐹 = 0𝑝)
46		df-0p 25578	. . . . . . . 8 ⊢ 0𝑝 = (ℂ × {0})
47	45, 46	eqtrdi 2781	. . . . . . 7 ⊢ (𝐹 = 0𝑝 → 𝐹 = (ℂ × {0}))
48	47	oveq1d 7405	. . . . . 6 ⊢ (𝐹 = 0𝑝 → (𝐹 ∘f · 𝐺) = ((ℂ × {0}) ∘f · 𝐺))
49	48	eqeq1d 2732	. . . . 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 26110	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
52	51	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
53		mul01 11360	. . . . . . 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 7691	. . . . 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 2781	. . . . . . 7 ⊢ (𝐺 = 0𝑝 → 𝐺 = (ℂ × {0}))
58	57	oveq2d 7406	. . . . . 6 ⊢ (𝐺 = 0𝑝 → (𝐹 ∘f · 𝐺) = (𝐹 ∘f · (ℂ × {0})))
59	58	eqeq1d 2732	. . . . 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 2743	. . 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 1540   ∈ wcel 2109  Vcvv 3450  {csn 4592   × cxp 5639  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∘_f cof 7654  ℂcc 11073  0cc0 11075   + caddc 11078   · cmul 11080  ℕ₀cn0 12449  0_𝑝c0p 25577  Polycply 26096  coeffccoe 26098  degcdgr 26099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-fl 13761  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-rlim 15462  df-sum 15660  df-0p 25578  df-ply 26100  df-coe 26102  df-dgr 26103

This theorem is referenced by:  plydiveu  26213  quotcan  26224  vieta1lem1  26225  vieta1lem2  26226