Theorem coe1mul3 23859

Description: The coefficient vector of multiplication in the univariate polynomial ring, at indices high enough that at most one component can be active in the sum. (Contributed by Stefan O'Rear, 25-Mar-2015.)

Hypotheses

Ref	Expression
coe1mul3.s	|- Y = (Poly1 ` R )
coe1mul3.t	|- .xb = ( .r ` Y )
coe1mul3.u	|- .x. = ( .r ` R )
coe1mul3.b	|- B = ( Base ` Y )
coe1mul3.d	|- D = ( deg1 ` R )
coe1mul3.r	|- ( ph -> R e. Ring )
coe1mul3.f1	|- ( ph -> F e. B )
coe1mul3.f2	|- ( ph -> I e. NN0 )
coe1mul3.f3	|- ( ph -> ( D ` F ) <_ I )
coe1mul3.g1	|- ( ph -> G e. B )
coe1mul3.g2	|- ( ph -> J e. NN0 )
coe1mul3.g3	|- ( ph -> ( D ` G ) <_ J )

Assertion

Ref	Expression
coe1mul3	|- ( ph -> ( (coe1 ` ( F .xb G ) ) ` ( I + J ) ) = ( ( (coe1 ` F ) ` I ) .x. ( (coe1 ` G ) ` J ) ) )

Proof of Theorem coe1mul3

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coe1mul3.r	. . . 4 |- ( ph -> R e. Ring )
2		coe1mul3.f1	. . . 4 |- ( ph -> F e. B )
3		coe1mul3.g1	. . . 4 |- ( ph -> G e. B )
4		coe1mul3.s	. . . . 5 |- Y = (Poly1 ` R )
5		coe1mul3.t	. . . . 5 |- .xb = ( .r ` Y )
6		coe1mul3.u	. . . . 5 |- .x. = ( .r ` R )
7		coe1mul3.b	. . . . 5 |- B = ( Base ` Y )
8	4, 5, 6, 7	coe1mul 19640	. . . 4 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> (coe1 ` ( F .xb G ) ) = ( x e. NN0 |-> ( R gsumg ( y e. ( 0 ... x ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( x - y ) ) ) ) ) ) )
9	1, 2, 3, 8	syl3anc 1326	. . 3 |- ( ph -> (coe1 ` ( F .xb G ) ) = ( x e. NN0 |-> ( R gsumg ( y e. ( 0 ... x ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( x - y ) ) ) ) ) ) )
10	9	fveq1d 6193	. 2 |- ( ph -> ( (coe1 ` ( F .xb G ) ) ` ( I + J ) ) = ( ( x e. NN0 |-> ( R gsumg ( y e. ( 0 ... x ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( x - y ) ) ) ) ) ) ` ( I + J ) ) )
11		coe1mul3.f2	. . . 4 |- ( ph -> I e. NN0 )
12		coe1mul3.g2	. . . 4 |- ( ph -> J e. NN0 )
13	11, 12	nn0addcld 11355	. . 3 |- ( ph -> ( I + J ) e. NN0 )
14		oveq2 6658	. . . . . 6 |- ( x = ( I + J ) -> ( 0 ... x ) = ( 0 ... ( I + J ) ) )
15		oveq1 6657	. . . . . . . 8 |- ( x = ( I + J ) -> ( x - y ) = ( ( I + J ) - y ) )
16	15	fveq2d 6195	. . . . . . 7 |- ( x = ( I + J ) -> ( (coe1 ` G ) ` ( x - y ) ) = ( (coe1 ` G ) ` ( ( I + J ) - y ) ) )
17	16	oveq2d 6666	. . . . . 6 |- ( x = ( I + J ) -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( x - y ) ) ) = ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) )
18	14, 17	mpteq12dv 4733	. . . . 5 |- ( x = ( I + J ) -> ( y e. ( 0 ... x ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( x - y ) ) ) ) = ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) )
19	18	oveq2d 6666	. . . 4 |- ( x = ( I + J ) -> ( R gsumg ( y e. ( 0 ... x ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( x - y ) ) ) ) ) = ( R gsumg ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) )
20		eqid 2622	. . . 4 |- ( x e. NN0 |-> ( R gsumg ( y e. ( 0 ... x ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( x - y ) ) ) ) ) ) = ( x e. NN0 |-> ( R gsumg ( y e. ( 0 ... x ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( x - y ) ) ) ) ) )
21		ovex 6678	. . . 4 |- ( R gsumg ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) e. _V
22	19, 20, 21	fvmpt 6282	. . 3 |- ( ( I + J ) e. NN0 -> ( ( x e. NN0 |-> ( R gsumg ( y e. ( 0 ... x ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( x - y ) ) ) ) ) ) ` ( I + J ) ) = ( R gsumg ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) )
23	13, 22	syl 17	. 2 |- ( ph -> ( ( x e. NN0 |-> ( R gsumg ( y e. ( 0 ... x ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( x - y ) ) ) ) ) ) ` ( I + J ) ) = ( R gsumg ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) )
24		eqid 2622	. . . 4 |- ( Base ` R ) = ( Base ` R )
25		eqid 2622	. . . 4 |- ( 0g ` R ) = ( 0g ` R )
26		ringmnd 18556	. . . . 5 |- ( R e. Ring -> R e. Mnd )
27	1, 26	syl 17	. . . 4 |- ( ph -> R e. Mnd )
28		ovexd 6680	. . . 4 |- ( ph -> ( 0 ... ( I + J ) ) e. _V )
29	11	nn0red 11352	. . . . . 6 |- ( ph -> I e. RR )
30		nn0addge1 11339	. . . . . 6 |- ( ( I e. RR /\ J e. NN0 ) -> I <_ ( I + J ) )
31	29, 12, 30	syl2anc 693	. . . . 5 |- ( ph -> I <_ ( I + J ) )
32		fznn0 12432	. . . . . 6 |- ( ( I + J ) e. NN0 -> ( I e. ( 0 ... ( I + J ) ) <-> ( I e. NN0 /\ I <_ ( I + J ) ) ) )
33	13, 32	syl 17	. . . . 5 |- ( ph -> ( I e. ( 0 ... ( I + J ) ) <-> ( I e. NN0 /\ I <_ ( I + J ) ) ) )
34	11, 31, 33	mpbir2and 957	. . . 4 |- ( ph -> I e. ( 0 ... ( I + J ) ) )
35	1	adantr 481	. . . . . 6 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> R e. Ring )
36		eqid 2622	. . . . . . . . 9 |- (coe1 ` F ) = (coe1 ` F )
37	36, 7, 4, 24	coe1f 19581	. . . . . . . 8 |- ( F e. B -> (coe1 ` F ) : NN0 --> ( Base ` R ) )
38	2, 37	syl 17	. . . . . . 7 |- ( ph -> (coe1 ` F ) : NN0 --> ( Base ` R ) )
39		elfznn0 12433	. . . . . . 7 |- ( y e. ( 0 ... ( I + J ) ) -> y e. NN0 )
40		ffvelrn 6357	. . . . . . 7 |- ( ( (coe1 ` F ) : NN0 --> ( Base ` R ) /\ y e. NN0 ) -> ( (coe1 ` F ) ` y ) e. ( Base ` R ) )
41	38, 39, 40	syl2an 494	. . . . . 6 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( (coe1 ` F ) ` y ) e. ( Base ` R ) )
42		eqid 2622	. . . . . . . . 9 |- (coe1 ` G ) = (coe1 ` G )
43	42, 7, 4, 24	coe1f 19581	. . . . . . . 8 |- ( G e. B -> (coe1 ` G ) : NN0 --> ( Base ` R ) )
44	3, 43	syl 17	. . . . . . 7 |- ( ph -> (coe1 ` G ) : NN0 --> ( Base ` R ) )
45		fznn0sub 12373	. . . . . . 7 |- ( y e. ( 0 ... ( I + J ) ) -> ( ( I + J ) - y ) e. NN0 )
46		ffvelrn 6357	. . . . . . 7 |- ( ( (coe1 ` G ) : NN0 --> ( Base ` R ) /\ ( ( I + J ) - y ) e. NN0 ) -> ( (coe1 ` G ) ` ( ( I + J ) - y ) ) e. ( Base ` R ) )
47	44, 45, 46	syl2an 494	. . . . . 6 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( (coe1 ` G ) ` ( ( I + J ) - y ) ) e. ( Base ` R ) )
48	24, 6	ringcl 18561	. . . . . 6 |- ( ( R e. Ring /\ ( (coe1 ` F ) ` y ) e. ( Base ` R ) /\ ( (coe1 ` G ) ` ( ( I + J ) - y ) ) e. ( Base ` R ) ) -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) e. ( Base ` R ) )
49	35, 41, 47, 48	syl3anc 1326	. . . . 5 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) e. ( Base ` R ) )
50		eqid 2622	. . . . 5 |- ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) = ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) )
51	49, 50	fmptd 6385	. . . 4 |- ( ph -> ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) : ( 0 ... ( I + J ) ) --> ( Base ` R ) )
52		eldifsn 4317	. . . . . 6 |- ( y e. ( ( 0 ... ( I + J ) ) \ { I } ) <-> ( y e. ( 0 ... ( I + J ) ) /\ y =/= I ) )
53	39	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> y e. NN0 )
54	53	nn0red 11352	. . . . . . . . 9 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> y e. RR )
55	29	adantr 481	. . . . . . . . 9 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> I e. RR )
56	54, 55	lttri2d 10176	. . . . . . . 8 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( y =/= I <-> ( y < I \/ I < y ) ) )
57	3	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> G e. B )
58	45	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( I + J ) - y ) e. NN0 )
59	58	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( ( I + J ) - y ) e. NN0 )
60		coe1mul3.d	. . . . . . . . . . . . . . . . 17 |- D = ( deg1 ` R )
61	60, 4, 7	deg1xrcl 23842	. . . . . . . . . . . . . . . 16 |- ( G e. B -> ( D ` G ) e. RR* )
62	3, 61	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( D ` G ) e. RR* )
63	62	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( D ` G ) e. RR* )
64	12	nn0red 11352	. . . . . . . . . . . . . . . 16 |- ( ph -> J e. RR )
65	64	rexrd 10089	. . . . . . . . . . . . . . 15 |- ( ph -> J e. RR* )
66	65	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> J e. RR* )
67	13	nn0red 11352	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( I + J ) e. RR )
68	67	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( I + J ) e. RR )
69	68, 54	resubcld 10458	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( I + J ) - y ) e. RR )
70	69	rexrd 10089	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( I + J ) - y ) e. RR* )
71	70	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( ( I + J ) - y ) e. RR* )
72		coe1mul3.g3	. . . . . . . . . . . . . . 15 |- ( ph -> ( D ` G ) <_ J )
73	72	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( D ` G ) <_ J )
74	64	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> J e. RR )
75	54, 55, 74	ltadd1d 10620	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( y < I <-> ( y + J ) < ( I + J ) ) )
76	54, 74, 68	ltaddsub2d 10628	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( y + J ) < ( I + J ) <-> J < ( ( I + J ) - y ) ) )
77	75, 76	bitrd 268	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( y < I <-> J < ( ( I + J ) - y ) ) )
78	77	biimpa 501	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> J < ( ( I + J ) - y ) )
79	63, 66, 71, 73, 78	xrlelttrd 11991	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( D ` G ) < ( ( I + J ) - y ) )
80	60, 4, 7, 25, 42	deg1lt 23857	. . . . . . . . . . . . 13 |- ( ( G e. B /\ ( ( I + J ) - y ) e. NN0 /\ ( D ` G ) < ( ( I + J ) - y ) ) -> ( (coe1 ` G ) ` ( ( I + J ) - y ) ) = ( 0g ` R ) )
81	57, 59, 79, 80	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( (coe1 ` G ) ` ( ( I + J ) - y ) ) = ( 0g ` R ) )
82	81	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( ( (coe1 ` F ) ` y ) .x. ( 0g ` R ) ) )
83	24, 6, 25	ringrz 18588	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ ( (coe1 ` F ) ` y ) e. ( Base ` R ) ) -> ( ( (coe1 ` F ) ` y ) .x. ( 0g ` R ) ) = ( 0g ` R ) )
84	35, 41, 83	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( (coe1 ` F ) ` y ) .x. ( 0g ` R ) ) = ( 0g ` R ) )
85	84	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( ( (coe1 ` F ) ` y ) .x. ( 0g ` R ) ) = ( 0g ` R ) )
86	82, 85	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) )
87	2	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> F e. B )
88	53	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> y e. NN0 )
89	60, 4, 7	deg1xrcl 23842	. . . . . . . . . . . . . . . 16 |- ( F e. B -> ( D ` F ) e. RR* )
90	2, 89	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( D ` F ) e. RR* )
91	90	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( D ` F ) e. RR* )
92	29	rexrd 10089	. . . . . . . . . . . . . . 15 |- ( ph -> I e. RR* )
93	92	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> I e. RR* )
94	54	rexrd 10089	. . . . . . . . . . . . . . 15 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> y e. RR* )
95	94	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> y e. RR* )
96		coe1mul3.f3	. . . . . . . . . . . . . . 15 |- ( ph -> ( D ` F ) <_ I )
97	96	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( D ` F ) <_ I )
98		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> I < y )
99	91, 93, 95, 97, 98	xrlelttrd 11991	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( D ` F ) < y )
100	60, 4, 7, 25, 36	deg1lt 23857	. . . . . . . . . . . . 13 |- ( ( F e. B /\ y e. NN0 /\ ( D ` F ) < y ) -> ( (coe1 ` F ) ` y ) = ( 0g ` R ) )
101	87, 88, 99, 100	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( (coe1 ` F ) ` y ) = ( 0g ` R ) )
102	101	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( ( 0g ` R ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) )
103	24, 6, 25	ringlz 18587	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ ( (coe1 ` G ) ` ( ( I + J ) - y ) ) e. ( Base ` R ) ) -> ( ( 0g ` R ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) )
104	35, 47, 103	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( 0g ` R ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) )
105	104	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( ( 0g ` R ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) )
106	102, 105	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) )
107	86, 106	jaodan 826	. . . . . . . . 9 |- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ ( y < I \/ I < y ) ) -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) )
108	107	ex 450	. . . . . . . 8 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( y < I \/ I < y ) -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) ) )
109	56, 108	sylbid 230	. . . . . . 7 |- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( y =/= I -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) ) )
110	109	impr 649	. . . . . 6 |- ( ( ph /\ ( y e. ( 0 ... ( I + J ) ) /\ y =/= I ) ) -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) )
111	52, 110	sylan2b 492	. . . . 5 |- ( ( ph /\ y e. ( ( 0 ... ( I + J ) ) \ { I } ) ) -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) )
112	111, 28	suppss2 7329	. . . 4 |- ( ph -> ( ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) supp ( 0g ` R ) ) C_ { I } )
113	24, 25, 27, 28, 34, 51, 112	gsumpt 18361	. . 3 |- ( ph -> ( R gsumg ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) = ( ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ` I ) )
114		fveq2 6191	. . . . . 6 |- ( y = I -> ( (coe1 ` F ) ` y ) = ( (coe1 ` F ) ` I ) )
115		oveq2 6658	. . . . . . 7 |- ( y = I -> ( ( I + J ) - y ) = ( ( I + J ) - I ) )
116	115	fveq2d 6195	. . . . . 6 |- ( y = I -> ( (coe1 ` G ) ` ( ( I + J ) - y ) ) = ( (coe1 ` G ) ` ( ( I + J ) - I ) ) )
117	114, 116	oveq12d 6668	. . . . 5 |- ( y = I -> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( ( (coe1 ` F ) ` I ) .x. ( (coe1 ` G ) ` ( ( I + J ) - I ) ) ) )
118		ovex 6678	. . . . 5 |- ( ( (coe1 ` F ) ` I ) .x. ( (coe1 ` G ) ` ( ( I + J ) - I ) ) ) e. _V
119	117, 50, 118	fvmpt 6282	. . . 4 |- ( I e. ( 0 ... ( I + J ) ) -> ( ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ` I ) = ( ( (coe1 ` F ) ` I ) .x. ( (coe1 ` G ) ` ( ( I + J ) - I ) ) ) )
120	34, 119	syl 17	. . 3 |- ( ph -> ( ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ` I ) = ( ( (coe1 ` F ) ` I ) .x. ( (coe1 ` G ) ` ( ( I + J ) - I ) ) ) )
121	11	nn0cnd 11353	. . . . . 6 |- ( ph -> I e. CC )
122	12	nn0cnd 11353	. . . . . 6 |- ( ph -> J e. CC )
123	121, 122	pncan2d 10394	. . . . 5 |- ( ph -> ( ( I + J ) - I ) = J )
124	123	fveq2d 6195	. . . 4 |- ( ph -> ( (coe1 ` G ) ` ( ( I + J ) - I ) ) = ( (coe1 ` G ) ` J ) )
125	124	oveq2d 6666	. . 3 |- ( ph -> ( ( (coe1 ` F ) ` I ) .x. ( (coe1 ` G ) ` ( ( I + J ) - I ) ) ) = ( ( (coe1 ` F ) ` I ) .x. ( (coe1 ` G ) ` J ) ) )
126	113, 120, 125	3eqtrd 2660	. 2 |- ( ph -> ( R gsumg ( y e. ( 0 ... ( I + J ) ) |-> ( ( (coe1 ` F ) ` y ) .x. ( (coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) = ( ( (coe1 ` F ) ` I ) .x. ( (coe1 ` G ) ` J ) ) )
127	10, 23, 126	3eqtrd 2660	1 |- ( ph -> ( (coe1 ` ( F .xb G ) ) ` ( I + J ) ) = ( ( (coe1 ` F ) ` I ) .x. ( (coe1 ` G ) ` J ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   {csn 4177   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NN0cn0 11292   ...cfz 12326   Basecbs 15857   .rcmulr 15942   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   Ringcrg 18547  Poly₁cpl1 19547  coe₁cco1 19548   deg₁ cdg1 23814

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-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-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  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-mulg 17541  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-psr 19356  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-ply1 19552  df-coe1 19553  df-cnfld 19747  df-mdeg 23815  df-deg1 23816

This theorem is referenced by:  coe1mul4  23860