Theorem ply1rem 23923

Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15260). If a polynomial F is divided by the linear factor x - A, the remainder is equal to F ( A ), the evaluation of the polynomial at A (interpreted as a constant polynomial). (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
ply1rem.p	|- P = (Poly1 ` R )
ply1rem.b	|- B = ( Base ` P )
ply1rem.k	|- K = ( Base ` R )
ply1rem.x	|- X = (var1 ` R )
ply1rem.m	|- .- = ( -g ` P )
ply1rem.a	|- A = (algSc ` P )
ply1rem.g	|- G = ( X .- ( A ` N ) )
ply1rem.o	|- O = (eval1 ` R )
ply1rem.1	|- ( ph -> R e. NzRing )
ply1rem.2	|- ( ph -> R e. CRing )
ply1rem.3	|- ( ph -> N e. K )
ply1rem.4	|- ( ph -> F e. B )
ply1rem.e	|- E = (rem1p ` R )

Assertion

Ref	Expression
ply1rem	|- ( ph -> ( F E G ) = ( A ` ( ( O ` F ) ` N ) ) )

Proof of Theorem ply1rem

Step	Hyp	Ref	Expression
1		ply1rem.1	. . . . . . . . 9 |- ( ph -> R e. NzRing )
2		nzrring 19261	. . . . . . . . 9 |- ( R e. NzRing -> R e. Ring )
3	1, 2	syl 17	. . . . . . . 8 |- ( ph -> R e. Ring )
4		ply1rem.4	. . . . . . . 8 |- ( ph -> F e. B )
5		ply1rem.p	. . . . . . . . . . 11 |- P = (Poly1 ` R )
6		ply1rem.b	. . . . . . . . . . 11 |- B = ( Base ` P )
7		ply1rem.k	. . . . . . . . . . 11 |- K = ( Base ` R )
8		ply1rem.x	. . . . . . . . . . 11 |- X = (var1 ` R )
9		ply1rem.m	. . . . . . . . . . 11 |- .- = ( -g ` P )
10		ply1rem.a	. . . . . . . . . . 11 |- A = (algSc ` P )
11		ply1rem.g	. . . . . . . . . . 11 |- G = ( X .- ( A ` N ) )
12		ply1rem.o	. . . . . . . . . . 11 |- O = (eval1 ` R )
13		ply1rem.2	. . . . . . . . . . 11 |- ( ph -> R e. CRing )
14		ply1rem.3	. . . . . . . . . . 11 |- ( ph -> N e. K )
15		eqid 2622	. . . . . . . . . . 11 |- (Monic1p ` R ) = (Monic1p ` R )
16		eqid 2622	. . . . . . . . . . 11 |- ( deg1 ` R ) = ( deg1 ` R )
17		eqid 2622	. . . . . . . . . . 11 |- ( 0g ` R ) = ( 0g ` R )
18	5, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17	ply1remlem 23922	. . . . . . . . . 10 |- ( ph -> ( G e. (Monic1p ` R ) /\ ( ( deg1 ` R ) ` G ) = 1 /\ ( `' ( O ` G ) " { ( 0g ` R ) } ) = { N } ) )
19	18	simp1d 1073	. . . . . . . . 9 |- ( ph -> G e. (Monic1p ` R ) )
20		eqid 2622	. . . . . . . . . 10 |- (Unic1p ` R ) = (Unic1p ` R )
21	20, 15	mon1puc1p 23910	. . . . . . . . 9 |- ( ( R e. Ring /\ G e. (Monic1p ` R ) ) -> G e. (Unic1p ` R ) )
22	3, 19, 21	syl2anc 693	. . . . . . . 8 |- ( ph -> G e. (Unic1p ` R ) )
23		ply1rem.e	. . . . . . . . 9 |- E = (rem1p ` R )
24	23, 5, 6, 20, 16	r1pdeglt 23918	. . . . . . . 8 |- ( ( R e. Ring /\ F e. B /\ G e. (Unic1p ` R ) ) -> ( ( deg1 ` R ) ` ( F E G ) ) < ( ( deg1 ` R ) ` G ) )
25	3, 4, 22, 24	syl3anc 1326	. . . . . . 7 |- ( ph -> ( ( deg1 ` R ) ` ( F E G ) ) < ( ( deg1 ` R ) ` G ) )
26	18	simp2d 1074	. . . . . . 7 |- ( ph -> ( ( deg1 ` R ) ` G ) = 1 )
27	25, 26	breqtrd 4679	. . . . . 6 |- ( ph -> ( ( deg1 ` R ) ` ( F E G ) ) < 1 )
28		1e0p1 11552	. . . . . 6 |- 1 = ( 0 + 1 )
29	27, 28	syl6breq 4694	. . . . 5 |- ( ph -> ( ( deg1 ` R ) ` ( F E G ) ) < ( 0 + 1 ) )
30		0nn0 11307	. . . . . 6 |- 0 e. NN0
31		nn0leltp1 11436	. . . . . 6 |- ( ( ( ( deg1 ` R ) ` ( F E G ) ) e. NN0 /\ 0 e. NN0 ) -> ( ( ( deg1 ` R ) ` ( F E G ) ) <_ 0 <-> ( ( deg1 ` R ) ` ( F E G ) ) < ( 0 + 1 ) ) )
32	30, 31	mpan2 707	. . . . 5 |- ( ( ( deg1 ` R ) ` ( F E G ) ) e. NN0 -> ( ( ( deg1 ` R ) ` ( F E G ) ) <_ 0 <-> ( ( deg1 ` R ) ` ( F E G ) ) < ( 0 + 1 ) ) )
33	29, 32	syl5ibrcom 237	. . . 4 |- ( ph -> ( ( ( deg1 ` R ) ` ( F E G ) ) e. NN0 -> ( ( deg1 ` R ) ` ( F E G ) ) <_ 0 ) )
34		elsni 4194	. . . . . 6 |- ( ( ( deg1 ` R ) ` ( F E G ) ) e. { -oo } -> ( ( deg1 ` R ) ` ( F E G ) ) = -oo )
35		0xr 10086	. . . . . . 7 |- 0 e. RR*
36		mnfle 11969	. . . . . . 7 |- ( 0 e. RR* -> -oo <_ 0 )
37	35, 36	ax-mp 5	. . . . . 6 |- -oo <_ 0
38	34, 37	syl6eqbr 4692	. . . . 5 |- ( ( ( deg1 ` R ) ` ( F E G ) ) e. { -oo } -> ( ( deg1 ` R ) ` ( F E G ) ) <_ 0 )
39	38	a1i 11	. . . 4 |- ( ph -> ( ( ( deg1 ` R ) ` ( F E G ) ) e. { -oo } -> ( ( deg1 ` R ) ` ( F E G ) ) <_ 0 ) )
40	23, 5, 6, 20	r1pcl 23917	. . . . . . 7 |- ( ( R e. Ring /\ F e. B /\ G e. (Unic1p ` R ) ) -> ( F E G ) e. B )
41	3, 4, 22, 40	syl3anc 1326	. . . . . 6 |- ( ph -> ( F E G ) e. B )
42	16, 5, 6	deg1cl 23843	. . . . . 6 |- ( ( F E G ) e. B -> ( ( deg1 ` R ) ` ( F E G ) ) e. ( NN0 u. { -oo } ) )
43	41, 42	syl 17	. . . . 5 |- ( ph -> ( ( deg1 ` R ) ` ( F E G ) ) e. ( NN0 u. { -oo } ) )
44		elun 3753	. . . . 5 |- ( ( ( deg1 ` R ) ` ( F E G ) ) e. ( NN0 u. { -oo } ) <-> ( ( ( deg1 ` R ) ` ( F E G ) ) e. NN0 \/ ( ( deg1 ` R ) ` ( F E G ) ) e. { -oo } ) )
45	43, 44	sylib 208	. . . 4 |- ( ph -> ( ( ( deg1 ` R ) ` ( F E G ) ) e. NN0 \/ ( ( deg1 ` R ) ` ( F E G ) ) e. { -oo } ) )
46	33, 39, 45	mpjaod 396	. . 3 |- ( ph -> ( ( deg1 ` R ) ` ( F E G ) ) <_ 0 )
47	16, 5, 6, 10	deg1le0 23871	. . . 4 |- ( ( R e. Ring /\ ( F E G ) e. B ) -> ( ( ( deg1 ` R ) ` ( F E G ) ) <_ 0 <-> ( F E G ) = ( A ` ( (coe1 ` ( F E G ) ) ` 0 ) ) ) )
48	3, 41, 47	syl2anc 693	. . 3 |- ( ph -> ( ( ( deg1 ` R ) ` ( F E G ) ) <_ 0 <-> ( F E G ) = ( A ` ( (coe1 ` ( F E G ) ) ` 0 ) ) ) )
49	46, 48	mpbid 222	. 2 |- ( ph -> ( F E G ) = ( A ` ( (coe1 ` ( F E G ) ) ` 0 ) ) )
50		eqid 2622	. . . . . . . . 9 |- (quot1p ` R ) = (quot1p ` R )
51		eqid 2622	. . . . . . . . 9 |- ( .r ` P ) = ( .r ` P )
52		eqid 2622	. . . . . . . . 9 |- ( +g ` P ) = ( +g ` P )
53	5, 6, 20, 50, 23, 51, 52	r1pid 23919	. . . . . . . 8 |- ( ( R e. Ring /\ F e. B /\ G e. (Unic1p ` R ) ) -> F = ( ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ( +g ` P ) ( F E G ) ) )
54	3, 4, 22, 53	syl3anc 1326	. . . . . . 7 |- ( ph -> F = ( ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ( +g ` P ) ( F E G ) ) )
55	54	fveq2d 6195	. . . . . 6 |- ( ph -> ( O ` F ) = ( O ` ( ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ( +g ` P ) ( F E G ) ) ) )
56		eqid 2622	. . . . . . . . . 10 |- ( R ^s K ) = ( R ^s K )
57	12, 5, 56, 7	evl1rhm 19696	. . . . . . . . 9 |- ( R e. CRing -> O e. ( P RingHom ( R ^s K ) ) )
58	13, 57	syl 17	. . . . . . . 8 |- ( ph -> O e. ( P RingHom ( R ^s K ) ) )
59		rhmghm 18725	. . . . . . . 8 |- ( O e. ( P RingHom ( R ^s K ) ) -> O e. ( P GrpHom ( R ^s K ) ) )
60	58, 59	syl 17	. . . . . . 7 |- ( ph -> O e. ( P GrpHom ( R ^s K ) ) )
61	5	ply1ring 19618	. . . . . . . . 9 |- ( R e. Ring -> P e. Ring )
62	3, 61	syl 17	. . . . . . . 8 |- ( ph -> P e. Ring )
63	50, 5, 6, 20	q1pcl 23915	. . . . . . . . 9 |- ( ( R e. Ring /\ F e. B /\ G e. (Unic1p ` R ) ) -> ( F (quot1p ` R ) G ) e. B )
64	3, 4, 22, 63	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( F (quot1p ` R ) G ) e. B )
65	5, 6, 15	mon1pcl 23904	. . . . . . . . 9 |- ( G e. (Monic1p ` R ) -> G e. B )
66	19, 65	syl 17	. . . . . . . 8 |- ( ph -> G e. B )
67	6, 51	ringcl 18561	. . . . . . . 8 |- ( ( P e. Ring /\ ( F (quot1p ` R ) G ) e. B /\ G e. B ) -> ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) e. B )
68	62, 64, 66, 67	syl3anc 1326	. . . . . . 7 |- ( ph -> ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) e. B )
69		eqid 2622	. . . . . . . 8 |- ( +g ` ( R ^s K ) ) = ( +g ` ( R ^s K ) )
70	6, 52, 69	ghmlin 17665	. . . . . . 7 |- ( ( O e. ( P GrpHom ( R ^s K ) ) /\ ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) e. B /\ ( F E G ) e. B ) -> ( O ` ( ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ( +g ` P ) ( F E G ) ) ) = ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) ( +g ` ( R ^s K ) ) ( O ` ( F E G ) ) ) )
71	60, 68, 41, 70	syl3anc 1326	. . . . . 6 |- ( ph -> ( O ` ( ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ( +g ` P ) ( F E G ) ) ) = ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) ( +g ` ( R ^s K ) ) ( O ` ( F E G ) ) ) )
72		eqid 2622	. . . . . . 7 |- ( Base ` ( R ^s K ) ) = ( Base ` ( R ^s K ) )
73		fvex 6201	. . . . . . . . 9 |- ( Base ` R ) e. _V
74	7, 73	eqeltri 2697	. . . . . . . 8 |- K e. _V
75	74	a1i 11	. . . . . . 7 |- ( ph -> K e. _V )
76	6, 72	rhmf 18726	. . . . . . . . 9 |- ( O e. ( P RingHom ( R ^s K ) ) -> O : B --> ( Base ` ( R ^s K ) ) )
77	58, 76	syl 17	. . . . . . . 8 |- ( ph -> O : B --> ( Base ` ( R ^s K ) ) )
78	77, 68	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) e. ( Base ` ( R ^s K ) ) )
79	77, 41	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( O ` ( F E G ) ) e. ( Base ` ( R ^s K ) ) )
80		eqid 2622	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
81	56, 72, 1, 75, 78, 79, 80, 69	pwsplusgval 16150	. . . . . 6 |- ( ph -> ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) ( +g ` ( R ^s K ) ) ( O ` ( F E G ) ) ) = ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) oF ( +g ` R ) ( O ` ( F E G ) ) ) )
82	55, 71, 81	3eqtrd 2660	. . . . 5 |- ( ph -> ( O ` F ) = ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) oF ( +g ` R ) ( O ` ( F E G ) ) ) )
83	82	fveq1d 6193	. . . 4 |- ( ph -> ( ( O ` F ) ` N ) = ( ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) oF ( +g ` R ) ( O ` ( F E G ) ) ) ` N ) )
84	56, 7, 72, 1, 75, 78	pwselbas 16149	. . . . . . 7 |- ( ph -> ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) : K --> K )
85		ffn 6045	. . . . . . 7 |- ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) : K --> K -> ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) Fn K )
86	84, 85	syl 17	. . . . . 6 |- ( ph -> ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) Fn K )
87	56, 7, 72, 1, 75, 79	pwselbas 16149	. . . . . . 7 |- ( ph -> ( O ` ( F E G ) ) : K --> K )
88		ffn 6045	. . . . . . 7 |- ( ( O ` ( F E G ) ) : K --> K -> ( O ` ( F E G ) ) Fn K )
89	87, 88	syl 17	. . . . . 6 |- ( ph -> ( O ` ( F E G ) ) Fn K )
90		fnfvof 6911	. . . . . 6 |- ( ( ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) Fn K /\ ( O ` ( F E G ) ) Fn K ) /\ ( K e. _V /\ N e. K ) ) -> ( ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) oF ( +g ` R ) ( O ` ( F E G ) ) ) ` N ) = ( ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) ` N ) ( +g ` R ) ( ( O ` ( F E G ) ) ` N ) ) )
91	86, 89, 75, 14, 90	syl22anc 1327	. . . . 5 |- ( ph -> ( ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) oF ( +g ` R ) ( O ` ( F E G ) ) ) ` N ) = ( ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) ` N ) ( +g ` R ) ( ( O ` ( F E G ) ) ` N ) ) )
92		eqid 2622	. . . . . . . . . . 11 |- ( .r ` ( R ^s K ) ) = ( .r ` ( R ^s K ) )
93	6, 51, 92	rhmmul 18727	. . . . . . . . . 10 |- ( ( O e. ( P RingHom ( R ^s K ) ) /\ ( F (quot1p ` R ) G ) e. B /\ G e. B ) -> ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) = ( ( O ` ( F (quot1p ` R ) G ) ) ( .r ` ( R ^s K ) ) ( O ` G ) ) )
94	58, 64, 66, 93	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) = ( ( O ` ( F (quot1p ` R ) G ) ) ( .r ` ( R ^s K ) ) ( O ` G ) ) )
95	77, 64	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( O ` ( F (quot1p ` R ) G ) ) e. ( Base ` ( R ^s K ) ) )
96	77, 66	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( O ` G ) e. ( Base ` ( R ^s K ) ) )
97		eqid 2622	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
98	56, 72, 1, 75, 95, 96, 97, 92	pwsmulrval 16151	. . . . . . . . 9 |- ( ph -> ( ( O ` ( F (quot1p ` R ) G ) ) ( .r ` ( R ^s K ) ) ( O ` G ) ) = ( ( O ` ( F (quot1p ` R ) G ) ) oF ( .r ` R ) ( O ` G ) ) )
99	94, 98	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) = ( ( O ` ( F (quot1p ` R ) G ) ) oF ( .r ` R ) ( O ` G ) ) )
100	99	fveq1d 6193	. . . . . . 7 |- ( ph -> ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) ` N ) = ( ( ( O ` ( F (quot1p ` R ) G ) ) oF ( .r ` R ) ( O ` G ) ) ` N ) )
101	56, 7, 72, 1, 75, 95	pwselbas 16149	. . . . . . . . 9 |- ( ph -> ( O ` ( F (quot1p ` R ) G ) ) : K --> K )
102		ffn 6045	. . . . . . . . 9 |- ( ( O ` ( F (quot1p ` R ) G ) ) : K --> K -> ( O ` ( F (quot1p ` R ) G ) ) Fn K )
103	101, 102	syl 17	. . . . . . . 8 |- ( ph -> ( O ` ( F (quot1p ` R ) G ) ) Fn K )
104	56, 7, 72, 1, 75, 96	pwselbas 16149	. . . . . . . . 9 |- ( ph -> ( O ` G ) : K --> K )
105		ffn 6045	. . . . . . . . 9 |- ( ( O ` G ) : K --> K -> ( O ` G ) Fn K )
106	104, 105	syl 17	. . . . . . . 8 |- ( ph -> ( O ` G ) Fn K )
107		fnfvof 6911	. . . . . . . 8 |- ( ( ( ( O ` ( F (quot1p ` R ) G ) ) Fn K /\ ( O ` G ) Fn K ) /\ ( K e. _V /\ N e. K ) ) -> ( ( ( O ` ( F (quot1p ` R ) G ) ) oF ( .r ` R ) ( O ` G ) ) ` N ) = ( ( ( O ` ( F (quot1p ` R ) G ) ) ` N ) ( .r ` R ) ( ( O ` G ) ` N ) ) )
108	103, 106, 75, 14, 107	syl22anc 1327	. . . . . . 7 |- ( ph -> ( ( ( O ` ( F (quot1p ` R ) G ) ) oF ( .r ` R ) ( O ` G ) ) ` N ) = ( ( ( O ` ( F (quot1p ` R ) G ) ) ` N ) ( .r ` R ) ( ( O ` G ) ` N ) ) )
109		snidg 4206	. . . . . . . . . . . . 13 |- ( N e. K -> N e. { N } )
110	14, 109	syl 17	. . . . . . . . . . . 12 |- ( ph -> N e. { N } )
111	18	simp3d 1075	. . . . . . . . . . . 12 |- ( ph -> ( `' ( O ` G ) " { ( 0g ` R ) } ) = { N } )
112	110, 111	eleqtrrd 2704	. . . . . . . . . . 11 |- ( ph -> N e. ( `' ( O ` G ) " { ( 0g ` R ) } ) )
113		fniniseg 6338	. . . . . . . . . . . 12 |- ( ( O ` G ) Fn K -> ( N e. ( `' ( O ` G ) " { ( 0g ` R ) } ) <-> ( N e. K /\ ( ( O ` G ) ` N ) = ( 0g ` R ) ) ) )
114	106, 113	syl 17	. . . . . . . . . . 11 |- ( ph -> ( N e. ( `' ( O ` G ) " { ( 0g ` R ) } ) <-> ( N e. K /\ ( ( O ` G ) ` N ) = ( 0g ` R ) ) ) )
115	112, 114	mpbid 222	. . . . . . . . . 10 |- ( ph -> ( N e. K /\ ( ( O ` G ) ` N ) = ( 0g ` R ) ) )
116	115	simprd 479	. . . . . . . . 9 |- ( ph -> ( ( O ` G ) ` N ) = ( 0g ` R ) )
117	116	oveq2d 6666	. . . . . . . 8 |- ( ph -> ( ( ( O ` ( F (quot1p ` R ) G ) ) ` N ) ( .r ` R ) ( ( O ` G ) ` N ) ) = ( ( ( O ` ( F (quot1p ` R ) G ) ) ` N ) ( .r ` R ) ( 0g ` R ) ) )
118	101, 14	ffvelrnd 6360	. . . . . . . . 9 |- ( ph -> ( ( O ` ( F (quot1p ` R ) G ) ) ` N ) e. K )
119	7, 97, 17	ringrz 18588	. . . . . . . . 9 |- ( ( R e. Ring /\ ( ( O ` ( F (quot1p ` R ) G ) ) ` N ) e. K ) -> ( ( ( O ` ( F (quot1p ` R ) G ) ) ` N ) ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) )
120	3, 118, 119	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( ( O ` ( F (quot1p ` R ) G ) ) ` N ) ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) )
121	117, 120	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( ( O ` ( F (quot1p ` R ) G ) ) ` N ) ( .r ` R ) ( ( O ` G ) ` N ) ) = ( 0g ` R ) )
122	100, 108, 121	3eqtrd 2660	. . . . . 6 |- ( ph -> ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) ` N ) = ( 0g ` R ) )
123	122	oveq1d 6665	. . . . 5 |- ( ph -> ( ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) ` N ) ( +g ` R ) ( ( O ` ( F E G ) ) ` N ) ) = ( ( 0g ` R ) ( +g ` R ) ( ( O ` ( F E G ) ) ` N ) ) )
124		ringgrp 18552	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
125	3, 124	syl 17	. . . . . 6 |- ( ph -> R e. Grp )
126	87, 14	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( ( O ` ( F E G ) ) ` N ) e. K )
127	7, 80, 17	grplid 17452	. . . . . 6 |- ( ( R e. Grp /\ ( ( O ` ( F E G ) ) ` N ) e. K ) -> ( ( 0g ` R ) ( +g ` R ) ( ( O ` ( F E G ) ) ` N ) ) = ( ( O ` ( F E G ) ) ` N ) )
128	125, 126, 127	syl2anc 693	. . . . 5 |- ( ph -> ( ( 0g ` R ) ( +g ` R ) ( ( O ` ( F E G ) ) ` N ) ) = ( ( O ` ( F E G ) ) ` N ) )
129	91, 123, 128	3eqtrd 2660	. . . 4 |- ( ph -> ( ( ( O ` ( ( F (quot1p ` R ) G ) ( .r ` P ) G ) ) oF ( +g ` R ) ( O ` ( F E G ) ) ) ` N ) = ( ( O ` ( F E G ) ) ` N ) )
130	49	fveq2d 6195	. . . . . . 7 |- ( ph -> ( O ` ( F E G ) ) = ( O ` ( A ` ( (coe1 ` ( F E G ) ) ` 0 ) ) ) )
131		eqid 2622	. . . . . . . . . . 11 |- (coe1 ` ( F E G ) ) = (coe1 ` ( F E G ) )
132	131, 6, 5, 7	coe1f 19581	. . . . . . . . . 10 |- ( ( F E G ) e. B -> (coe1 ` ( F E G ) ) : NN0 --> K )
133	41, 132	syl 17	. . . . . . . . 9 |- ( ph -> (coe1 ` ( F E G ) ) : NN0 --> K )
134		ffvelrn 6357	. . . . . . . . 9 |- ( ( (coe1 ` ( F E G ) ) : NN0 --> K /\ 0 e. NN0 ) -> ( (coe1 ` ( F E G ) ) ` 0 ) e. K )
135	133, 30, 134	sylancl 694	. . . . . . . 8 |- ( ph -> ( (coe1 ` ( F E G ) ) ` 0 ) e. K )
136	12, 5, 7, 10	evl1sca 19698	. . . . . . . 8 |- ( ( R e. CRing /\ ( (coe1 ` ( F E G ) ) ` 0 ) e. K ) -> ( O ` ( A ` ( (coe1 ` ( F E G ) ) ` 0 ) ) ) = ( K X. { ( (coe1 ` ( F E G ) ) ` 0 ) } ) )
137	13, 135, 136	syl2anc 693	. . . . . . 7 |- ( ph -> ( O ` ( A ` ( (coe1 ` ( F E G ) ) ` 0 ) ) ) = ( K X. { ( (coe1 ` ( F E G ) ) ` 0 ) } ) )
138	130, 137	eqtrd 2656	. . . . . 6 |- ( ph -> ( O ` ( F E G ) ) = ( K X. { ( (coe1 ` ( F E G ) ) ` 0 ) } ) )
139	138	fveq1d 6193	. . . . 5 |- ( ph -> ( ( O ` ( F E G ) ) ` N ) = ( ( K X. { ( (coe1 ` ( F E G ) ) ` 0 ) } ) ` N ) )
140		fvex 6201	. . . . . . 7 |- ( (coe1 ` ( F E G ) ) ` 0 ) e. _V
141	140	fvconst2 6469	. . . . . 6 |- ( N e. K -> ( ( K X. { ( (coe1 ` ( F E G ) ) ` 0 ) } ) ` N ) = ( (coe1 ` ( F E G ) ) ` 0 ) )
142	14, 141	syl 17	. . . . 5 |- ( ph -> ( ( K X. { ( (coe1 ` ( F E G ) ) ` 0 ) } ) ` N ) = ( (coe1 ` ( F E G ) ) ` 0 ) )
143	139, 142	eqtrd 2656	. . . 4 |- ( ph -> ( ( O ` ( F E G ) ) ` N ) = ( (coe1 ` ( F E G ) ) ` 0 ) )
144	83, 129, 143	3eqtrd 2660	. . 3 |- ( ph -> ( ( O ` F ) ` N ) = ( (coe1 ` ( F E G ) ) ` 0 ) )
145	144	fveq2d 6195	. 2 |- ( ph -> ( A ` ( ( O ` F ) ` N ) ) = ( A ` ( (coe1 ` ( F E G ) ) ` 0 ) ) )
146	49, 145	eqtr4d 2659	1 |- ( ph -> ( F E G ) = ( A ` ( ( O ` F ) ` N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   {csn 4177   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   0cc0 9936   1c1 9937   + caddc 9939   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   NN0cn0 11292   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   ^_s cpws 16107   Grpcgrp 17422   -gcsg 17424   GrpHom cghm 17657   Ringcrg 18547   CRingccrg 18548   RingHom crh 18712  NzRingcnzr 19257  algSccascl 19311  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548  eval₁ce1 19679   deg₁ cdg1 23814  Monic_1pcmn1 23885  Unic_1pcuc1p 23886  quot_1pcq1p 23887  rem_1pcr1p 23888

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  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-iin 4523  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-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  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-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-nzr 19258  df-rlreg 19283  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-evls 19506  df-evl 19507  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-evl1 19681  df-cnfld 19747  df-mdeg 23815  df-deg1 23816  df-mon1 23890  df-uc1p 23891  df-q1p 23892  df-r1p 23893

This theorem is referenced by:  facth1  23924