Theorem plymul0or 24036

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

Assertion

Ref	Expression
plymul0or	|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( F oF x. 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 e. (Poly ` S ) -> (deg ` F ) e. NN0 )
2		dgrcl 23989	. . . . . . 7 |- ( G e. (Poly ` S ) -> (deg ` G ) e. NN0 )
3		nn0addcl 11328	. . . . . . 7 |- ( ( (deg ` F ) e. NN0 /\ (deg ` G ) e. NN0 ) -> ( (deg ` F ) + (deg ` G ) ) e. NN0 )
4	1, 2, 3	syl2an 494	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (deg ` F ) + (deg ` G ) ) e. NN0 )
5		c0ex 10034	. . . . . . 7 |- 0 e. _V
6	5	fvconst2 6469	. . . . . 6 |- ( ( (deg ` F ) + (deg ` G ) ) e. NN0 -> ( ( NN0 X. { 0 } ) ` ( (deg ` F ) + (deg ` G ) ) ) = 0 )
7	4, 6	syl 17	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( NN0 X. { 0 } ) ` ( (deg ` F ) + (deg ` G ) ) ) = 0 )
8		fveq2 6191	. . . . . . . 8 |- ( ( F oF x. G ) = 0p -> (coeff ` ( F oF x. G ) ) = (coeff ` 0p ) )
9		coe0 24012	. . . . . . . 8 |- (coeff ` 0p ) = ( NN0 X. { 0 } )
10	8, 9	syl6eq 2672	. . . . . . 7 |- ( ( F oF x. G ) = 0p -> (coeff ` ( F oF x. G ) ) = ( NN0 X. { 0 } ) )
11	10	fveq1d 6193	. . . . . 6 |- ( ( F oF x. G ) = 0p -> ( (coeff ` ( F oF x. G ) ) ` ( (deg ` F ) + (deg ` G ) ) ) = ( ( NN0 X. { 0 } ) ` ( (deg ` F ) + (deg ` G ) ) ) )
12	11	eqeq1d 2624	. . . . 5 |- ( ( F oF x. G ) = 0p -> ( ( (coeff ` ( F oF x. G ) ) ` ( (deg ` F ) + (deg ` G ) ) ) = 0 <-> ( ( NN0 X. { 0 } ) ` ( (deg ` F ) + (deg ` G ) ) ) = 0 ) )
13	7, 12	syl5ibrcom 237	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( F oF x. G ) = 0p -> ( (coeff ` ( F oF x. 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 e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (coeff ` ( F oF x. G ) ) ` ( (deg ` F ) + (deg ` G ) ) ) = ( ( (coeff ` F ) ` (deg ` F ) ) x. ( (coeff ` G ) ` (deg ` G ) ) ) )
19	18	eqeq1d 2624	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( (coeff ` ( F oF x. G ) ) ` ( (deg ` F ) + (deg ` G ) ) ) = 0 <-> ( ( (coeff ` F ) ` (deg ` F ) ) x. ( (coeff ` G ) ` (deg ` G ) ) ) = 0 ) )
20	14	coef3 23988	. . . . . . . 8 |- ( F e. (Poly ` S ) -> (coeff ` F ) : NN0 --> CC )
21	20	adantr 481	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (coeff ` F ) : NN0 --> CC )
22	1	adantr 481	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (deg ` F ) e. NN0 )
23	21, 22	ffvelrnd 6360	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (coeff ` F ) ` (deg ` F ) ) e. CC )
24	15	coef3 23988	. . . . . . . 8 |- ( G e. (Poly ` S ) -> (coeff ` G ) : NN0 --> CC )
25	24	adantl 482	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (coeff ` G ) : NN0 --> CC )
26	2	adantl 482	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (deg ` G ) e. NN0 )
27	25, 26	ffvelrnd 6360	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (coeff ` G ) ` (deg ` G ) ) e. CC )
28	23, 27	mul0ord 10677	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( ( (coeff ` F ) ` (deg ` F ) ) x. ( (coeff ` G ) ` (deg ` G ) ) ) = 0 <-> ( ( (coeff ` F ) ` (deg ` F ) ) = 0 \/ ( (coeff ` G ) ` (deg ` G ) ) = 0 ) ) )
29	19, 28	bitrd 268	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( (coeff ` ( F oF x. G ) ) ` ( (deg ` F ) + (deg ` G ) ) ) = 0 <-> ( ( (coeff ` F ) ` (deg ` F ) ) = 0 \/ ( (coeff ` G ) ` (deg ` G ) ) = 0 ) ) )
30	13, 29	sylibd 229	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( F oF x. G ) = 0p -> ( ( (coeff ` F ) ` (deg ` F ) ) = 0 \/ ( (coeff ` G ) ` (deg ` G ) ) = 0 ) ) )
31	16, 14	dgreq0 24021	. . . . 5 |- ( F e. (Poly ` S ) -> ( F = 0p <-> ( (coeff ` F ) ` (deg ` F ) ) = 0 ) )
32	31	adantr 481	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F = 0p <-> ( (coeff ` F ) ` (deg ` F ) ) = 0 ) )
33	17, 15	dgreq0 24021	. . . . 5 |- ( G e. (Poly ` S ) -> ( G = 0p <-> ( (coeff ` G ) ` (deg ` G ) ) = 0 ) )
34	33	adantl 482	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( G = 0p <-> ( (coeff ` G ) ` (deg ` G ) ) = 0 ) )
35	32, 34	orbi12d 746	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( F = 0p \/ G = 0p ) <-> ( ( (coeff ` F ) ` (deg ` F ) ) = 0 \/ ( (coeff ` G ) ` (deg ` G ) ) = 0 ) ) )
36	30, 35	sylibrd 249	. 2 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( F oF x. G ) = 0p -> ( F = 0p \/ G = 0p ) ) )
37		cnex 10017	. . . . . . 7 |- CC e. _V
38	37	a1i 11	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> CC e. _V )
39		plyf 23954	. . . . . . 7 |- ( G e. (Poly ` S ) -> G : CC --> CC )
40	39	adantl 482	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> G : CC --> CC )
41		0cnd 10033	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> 0 e. CC )
42		mul02 10214	. . . . . . 7 |- ( x e. CC -> ( 0 x. x ) = 0 )
43	42	adantl 482	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ x e. CC ) -> ( 0 x. x ) = 0 )
44	38, 40, 41, 41, 43	caofid2 6928	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( CC X. { 0 } ) oF x. G ) = ( CC X. { 0 } ) )
45		id 22	. . . . . . . 8 |- ( F = 0p -> F = 0p )
46		df-0p 23437	. . . . . . . 8 |- 0p = ( CC X. { 0 } )
47	45, 46	syl6eq 2672	. . . . . . 7 |- ( F = 0p -> F = ( CC X. { 0 } ) )
48	47	oveq1d 6665	. . . . . 6 |- ( F = 0p -> ( F oF x. G ) = ( ( CC X. { 0 } ) oF x. G ) )
49	48	eqeq1d 2624	. . . . 5 |- ( F = 0p -> ( ( F oF x. G ) = ( CC X. { 0 } ) <-> ( ( CC X. { 0 } ) oF x. G ) = ( CC X. { 0 } ) ) )
50	44, 49	syl5ibrcom 237	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F = 0p -> ( F oF x. G ) = ( CC X. { 0 } ) ) )
51		plyf 23954	. . . . . . 7 |- ( F e. (Poly ` S ) -> F : CC --> CC )
52	51	adantr 481	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> F : CC --> CC )
53		mul01 10215	. . . . . . 7 |- ( x e. CC -> ( x x. 0 ) = 0 )
54	53	adantl 482	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ x e. CC ) -> ( x x. 0 ) = 0 )
55	38, 52, 41, 41, 54	caofid1 6927	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F oF x. ( CC X. { 0 } ) ) = ( CC X. { 0 } ) )
56		id 22	. . . . . . . 8 |- ( G = 0p -> G = 0p )
57	56, 46	syl6eq 2672	. . . . . . 7 |- ( G = 0p -> G = ( CC X. { 0 } ) )
58	57	oveq2d 6666	. . . . . 6 |- ( G = 0p -> ( F oF x. G ) = ( F oF x. ( CC X. { 0 } ) ) )
59	58	eqeq1d 2624	. . . . 5 |- ( G = 0p -> ( ( F oF x. G ) = ( CC X. { 0 } ) <-> ( F oF x. ( CC X. { 0 } ) ) = ( CC X. { 0 } ) ) )
60	55, 59	syl5ibrcom 237	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( G = 0p -> ( F oF x. G ) = ( CC X. { 0 } ) ) )
61	50, 60	jaod 395	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( F = 0p \/ G = 0p ) -> ( F oF x. G ) = ( CC X. { 0 } ) ) )
62	46	eqeq2i 2634	. . 3 |- ( ( F oF x. G ) = 0p <-> ( F oF x. G ) = ( CC X. { 0 } ) )
63	61, 62	syl6ibr 242	. 2 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( F = 0p \/ G = 0p ) -> ( F oF x. G ) = 0p ) )
64	36, 63	impbid 202	1 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( F oF x. G ) = 0p <-> ( F = 0p \/ G = 0p ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   + caddc 9939   x. cmul 9941   NN0cn0 11292   0pc0p 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