Theorem plymul0or 24036

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 23989	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
2		dgrcl 23989	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
3		nn0addcl 11328	. . . . . . 7 ⊢ (((deg‘𝐹) ∈ ℕ0 ∧ (deg‘𝐺) ∈ ℕ0) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
4	1, 2, 3	syl2an 494	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((deg‘𝐹) + (deg‘𝐺)) ∈ ℕ0)
5		c0ex 10034	. . . . . . 7 ⊢ 0 ∈ V
6	5	fvconst2 6469	. . . . . 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 6191	. . . . . . . 8 ⊢ ((𝐹 ∘𝑓 · 𝐺) = 0𝑝 → (coeff‘(𝐹 ∘𝑓 · 𝐺)) = (coeff‘0𝑝))
9		coe0 24012	. . . . . . . 8 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
10	8, 9	syl6eq 2672	. . . . . . 7 ⊢ ((𝐹 ∘𝑓 · 𝐺) = 0𝑝 → (coeff‘(𝐹 ∘𝑓 · 𝐺)) = (ℕ0 × {0}))
11	10	fveq1d 6193	. . . . . 6 ⊢ ((𝐹 ∘𝑓 · 𝐺) = 0𝑝 → ((coeff‘(𝐹 ∘𝑓 · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = ((ℕ0 × {0})‘((deg‘𝐹) + (deg‘𝐺))))
12	11	eqeq1d 2624	. . . . 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 237	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘𝑓 · 𝐺) = 0𝑝 → ((coeff‘(𝐹 ∘𝑓 · 𝐺))‘((deg‘𝐹) + (deg‘𝐺))) = 0))
14		eqid 2622	. . . . . . 7 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
15		eqid 2622	. . . . . . 7 ⊢ (coeff‘𝐺) = (coeff‘𝐺)
16		eqid 2622	. . . . . . 7 ⊢ (deg‘𝐹) = (deg‘𝐹)
17		eqid 2622	. . . . . . 7 ⊢ (deg‘𝐺) = (deg‘𝐺)
18	14, 15, 16, 17	coemulhi 24010	. . . . . 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 2624	. . . . 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 23988	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
21	20	adantr 481	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
22	1	adantr 481	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℕ0)
23	21, 22	ffvelrnd 6360	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐹)‘(deg‘𝐹)) ∈ ℂ)
24	15	coef3 23988	. . . . . . . 8 ⊢ (𝐺 ∈ (Poly‘𝑆) → (coeff‘𝐺):ℕ0⟶ℂ)
25	24	adantl 482	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘𝐺):ℕ0⟶ℂ)
26	2	adantl 482	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘𝐺) ∈ ℕ0)
27	25, 26	ffvelrnd 6360	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘𝐺)‘(deg‘𝐺)) ∈ ℂ)
28	23, 27	mul0ord 10677	. . . . 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 268	. . . 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 229	. . 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 24021	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
32	31	adantr 481	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 ↔ ((coeff‘𝐹)‘(deg‘𝐹)) = 0))
33	17, 15	dgreq0 24021	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
34	33	adantl 482	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 ↔ ((coeff‘𝐺)‘(deg‘𝐺)) = 0))
35	32, 34	orbi12d 746	. . 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 249	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘𝑓 · 𝐺) = 0𝑝 → (𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝)))
37		cnex 10017	. . . . . . 7 ⊢ ℂ ∈ V
38	37	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ℂ ∈ V)
39		plyf 23954	. . . . . . 7 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
40	39	adantl 482	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺:ℂ⟶ℂ)
41		0cnd 10033	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ∈ ℂ)
42		mul02 10214	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
43	42	adantl 482	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0)
44	38, 40, 41, 41, 43	caofid2 6928	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((ℂ × {0}) ∘𝑓 · 𝐺) = (ℂ × {0}))
45		id 22	. . . . . . . 8 ⊢ (𝐹 = 0𝑝 → 𝐹 = 0𝑝)
46		df-0p 23437	. . . . . . . 8 ⊢ 0𝑝 = (ℂ × {0})
47	45, 46	syl6eq 2672	. . . . . . 7 ⊢ (𝐹 = 0𝑝 → 𝐹 = (ℂ × {0}))
48	47	oveq1d 6665	. . . . . 6 ⊢ (𝐹 = 0𝑝 → (𝐹 ∘𝑓 · 𝐺) = ((ℂ × {0}) ∘𝑓 · 𝐺))
49	48	eqeq1d 2624	. . . . 5 ⊢ (𝐹 = 0𝑝 → ((𝐹 ∘𝑓 · 𝐺) = (ℂ × {0}) ↔ ((ℂ × {0}) ∘𝑓 · 𝐺) = (ℂ × {0})))
50	44, 49	syl5ibrcom 237	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 0𝑝 → (𝐹 ∘𝑓 · 𝐺) = (ℂ × {0})))
51		plyf 23954	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
52	51	adantr 481	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
53		mul01 10215	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
54	53	adantl 482	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑥 ∈ ℂ) → (𝑥 · 0) = 0)
55	38, 52, 41, 41, 54	caofid1 6927	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 · (ℂ × {0})) = (ℂ × {0}))
56		id 22	. . . . . . . 8 ⊢ (𝐺 = 0𝑝 → 𝐺 = 0𝑝)
57	56, 46	syl6eq 2672	. . . . . . 7 ⊢ (𝐺 = 0𝑝 → 𝐺 = (ℂ × {0}))
58	57	oveq2d 6666	. . . . . 6 ⊢ (𝐺 = 0𝑝 → (𝐹 ∘𝑓 · 𝐺) = (𝐹 ∘𝑓 · (ℂ × {0})))
59	58	eqeq1d 2624	. . . . 5 ⊢ (𝐺 = 0𝑝 → ((𝐹 ∘𝑓 · 𝐺) = (ℂ × {0}) ↔ (𝐹 ∘𝑓 · (ℂ × {0})) = (ℂ × {0})))
60	55, 59	syl5ibrcom 237	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐺 = 0𝑝 → (𝐹 ∘𝑓 · 𝐺) = (ℂ × {0})))
61	50, 60	jaod 395	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝) → (𝐹 ∘𝑓 · 𝐺) = (ℂ × {0})))
62	46	eqeq2i 2634	. . 3 ⊢ ((𝐹 ∘𝑓 · 𝐺) = 0𝑝 ↔ (𝐹 ∘𝑓 · 𝐺) = (ℂ × {0}))
63	61, 62	syl6ibr 242	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝) → (𝐹 ∘𝑓 · 𝐺) = 0𝑝))
64	36, 63	impbid 202	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘𝑓 · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  ℂcc 9934  0cc0 9936   + caddc 9939   · cmul 9941  ℕ₀cn0 11292  0_𝑝c0p 23436  Polycply 23940  coeffccoe 23942  degcdgr 23943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  plydiveu  24053  quotcan  24064  vieta1lem1  24065  vieta1lem2  24066