Theorem mplcoe3 19466

Description: Decompose a monomial in one variable into a power of a variable. (Contributed by Mario Carneiro, 7-Jan-2015.) (Proof shortened by AV, 18-Jul-2019.)

Hypotheses

Ref	Expression
mplcoe1.p	|- P = ( I mPoly R )
mplcoe1.d	|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
mplcoe1.z	|- .0. = ( 0g ` R )
mplcoe1.o	|- .1. = ( 1r ` R )
mplcoe1.i	|- ( ph -> I e. W )
mplcoe2.g	|- G = (mulGrp ` P )
mplcoe2.m	|- .^ = (.g ` G )
mplcoe2.v	|- V = ( I mVar R )
mplcoe3.r	|- ( ph -> R e. Ring )
mplcoe3.x	|- ( ph -> X e. I )
mplcoe3.n	|- ( ph -> N e. NN0 )

Assertion

Ref	Expression
mplcoe3	|- ( ph -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , N , 0 ) ) , .1. , .0. ) ) = ( N .^ ( V ` X ) ) )

Distinct variable groups:   .^ , k   y, k, .1.   k, G   f, k, y, I   k, N, y   ph, k, y   R, f, y   D, k, y   P, k   k, V   .0. , f, k, y   f, X, k, y   k, W, y

Allowed substitution hints:   ph( f)   D( f)   P( y, f)   R( k)   .1. ( f)   .^ ( y, f)   G( y, f)   N( f)   V( y, f)   W( f)

Proof of Theorem mplcoe3

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

Step	Hyp	Ref	Expression
1		mplcoe3.n	. 2 |- ( ph -> N e. NN0 )
2		ifeq1 4090	. . . . . . . . . . 11 |- ( x = 0 -> if ( k = X , x , 0 ) = if ( k = X , 0 , 0 ) )
3		ifid 4125	. . . . . . . . . . 11 |- if ( k = X , 0 , 0 ) = 0
4	2, 3	syl6eq 2672	. . . . . . . . . 10 |- ( x = 0 -> if ( k = X , x , 0 ) = 0 )
5	4	mpteq2dv 4745	. . . . . . . . 9 |- ( x = 0 -> ( k e. I |-> if ( k = X , x , 0 ) ) = ( k e. I |-> 0 ) )
6		fconstmpt 5163	. . . . . . . . 9 |- ( I X. { 0 } ) = ( k e. I |-> 0 )
7	5, 6	syl6eqr 2674	. . . . . . . 8 |- ( x = 0 -> ( k e. I |-> if ( k = X , x , 0 ) ) = ( I X. { 0 } ) )
8	7	eqeq2d 2632	. . . . . . 7 |- ( x = 0 -> ( y = ( k e. I |-> if ( k = X , x , 0 ) ) <-> y = ( I X. { 0 } ) ) )
9	8	ifbid 4108	. . . . . 6 |- ( x = 0 -> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) = if ( y = ( I X. { 0 } ) , .1. , .0. ) )
10	9	mpteq2dv 4745	. . . . 5 |- ( x = 0 -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) ) = ( y e. D |-> if ( y = ( I X. { 0 } ) , .1. , .0. ) ) )
11		oveq1 6657	. . . . 5 |- ( x = 0 -> ( x .^ ( V ` X ) ) = ( 0 .^ ( V ` X ) ) )
12	10, 11	eqeq12d 2637	. . . 4 |- ( x = 0 -> ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) ) = ( x .^ ( V ` X ) ) <-> ( y e. D |-> if ( y = ( I X. { 0 } ) , .1. , .0. ) ) = ( 0 .^ ( V ` X ) ) ) )
13	12	imbi2d 330	. . 3 |- ( x = 0 -> ( ( ph -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) ) = ( x .^ ( V ` X ) ) ) <-> ( ph -> ( y e. D |-> if ( y = ( I X. { 0 } ) , .1. , .0. ) ) = ( 0 .^ ( V ` X ) ) ) ) )
14		ifeq1 4090	. . . . . . . . 9 |- ( x = n -> if ( k = X , x , 0 ) = if ( k = X , n , 0 ) )
15	14	mpteq2dv 4745	. . . . . . . 8 |- ( x = n -> ( k e. I |-> if ( k = X , x , 0 ) ) = ( k e. I |-> if ( k = X , n , 0 ) ) )
16	15	eqeq2d 2632	. . . . . . 7 |- ( x = n -> ( y = ( k e. I |-> if ( k = X , x , 0 ) ) <-> y = ( k e. I |-> if ( k = X , n , 0 ) ) ) )
17	16	ifbid 4108	. . . . . 6 |- ( x = n -> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) = if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) )
18	17	mpteq2dv 4745	. . . . 5 |- ( x = n -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) ) = ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) )
19		oveq1 6657	. . . . 5 |- ( x = n -> ( x .^ ( V ` X ) ) = ( n .^ ( V ` X ) ) )
20	18, 19	eqeq12d 2637	. . . 4 |- ( x = n -> ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) ) = ( x .^ ( V ` X ) ) <-> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) = ( n .^ ( V ` X ) ) ) )
21	20	imbi2d 330	. . 3 |- ( x = n -> ( ( ph -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) ) = ( x .^ ( V ` X ) ) ) <-> ( ph -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) = ( n .^ ( V ` X ) ) ) ) )
22		ifeq1 4090	. . . . . . . . 9 |- ( x = ( n + 1 ) -> if ( k = X , x , 0 ) = if ( k = X , ( n + 1 ) , 0 ) )
23	22	mpteq2dv 4745	. . . . . . . 8 |- ( x = ( n + 1 ) -> ( k e. I |-> if ( k = X , x , 0 ) ) = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) )
24	23	eqeq2d 2632	. . . . . . 7 |- ( x = ( n + 1 ) -> ( y = ( k e. I |-> if ( k = X , x , 0 ) ) <-> y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) ) )
25	24	ifbid 4108	. . . . . 6 |- ( x = ( n + 1 ) -> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) = if ( y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) , .1. , .0. ) )
26	25	mpteq2dv 4745	. . . . 5 |- ( x = ( n + 1 ) -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) ) = ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) , .1. , .0. ) ) )
27		oveq1 6657	. . . . 5 |- ( x = ( n + 1 ) -> ( x .^ ( V ` X ) ) = ( ( n + 1 ) .^ ( V ` X ) ) )
28	26, 27	eqeq12d 2637	. . . 4 |- ( x = ( n + 1 ) -> ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) ) = ( x .^ ( V ` X ) ) <-> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) , .1. , .0. ) ) = ( ( n + 1 ) .^ ( V ` X ) ) ) )
29	28	imbi2d 330	. . 3 |- ( x = ( n + 1 ) -> ( ( ph -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) ) = ( x .^ ( V ` X ) ) ) <-> ( ph -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) , .1. , .0. ) ) = ( ( n + 1 ) .^ ( V ` X ) ) ) ) )
30		ifeq1 4090	. . . . . . . . 9 |- ( x = N -> if ( k = X , x , 0 ) = if ( k = X , N , 0 ) )
31	30	mpteq2dv 4745	. . . . . . . 8 |- ( x = N -> ( k e. I |-> if ( k = X , x , 0 ) ) = ( k e. I |-> if ( k = X , N , 0 ) ) )
32	31	eqeq2d 2632	. . . . . . 7 |- ( x = N -> ( y = ( k e. I |-> if ( k = X , x , 0 ) ) <-> y = ( k e. I |-> if ( k = X , N , 0 ) ) ) )
33	32	ifbid 4108	. . . . . 6 |- ( x = N -> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) = if ( y = ( k e. I |-> if ( k = X , N , 0 ) ) , .1. , .0. ) )
34	33	mpteq2dv 4745	. . . . 5 |- ( x = N -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) ) = ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , N , 0 ) ) , .1. , .0. ) ) )
35		oveq1 6657	. . . . 5 |- ( x = N -> ( x .^ ( V ` X ) ) = ( N .^ ( V ` X ) ) )
36	34, 35	eqeq12d 2637	. . . 4 |- ( x = N -> ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) ) = ( x .^ ( V ` X ) ) <-> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , N , 0 ) ) , .1. , .0. ) ) = ( N .^ ( V ` X ) ) ) )
37	36	imbi2d 330	. . 3 |- ( x = N -> ( ( ph -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , x , 0 ) ) , .1. , .0. ) ) = ( x .^ ( V ` X ) ) ) <-> ( ph -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , N , 0 ) ) , .1. , .0. ) ) = ( N .^ ( V ` X ) ) ) ) )
38		mplcoe1.p	. . . . . 6 |- P = ( I mPoly R )
39		mplcoe2.v	. . . . . 6 |- V = ( I mVar R )
40		eqid 2622	. . . . . 6 |- ( Base ` P ) = ( Base ` P )
41		mplcoe1.i	. . . . . 6 |- ( ph -> I e. W )
42		mplcoe3.r	. . . . . 6 |- ( ph -> R e. Ring )
43		mplcoe3.x	. . . . . 6 |- ( ph -> X e. I )
44	38, 39, 40, 41, 42, 43	mvrcl 19449	. . . . 5 |- ( ph -> ( V ` X ) e. ( Base ` P ) )
45		mplcoe2.g	. . . . . . 7 |- G = (mulGrp ` P )
46	45, 40	mgpbas 18495	. . . . . 6 |- ( Base ` P ) = ( Base ` G )
47		eqid 2622	. . . . . . 7 |- ( 1r ` P ) = ( 1r ` P )
48	45, 47	ringidval 18503	. . . . . 6 |- ( 1r ` P ) = ( 0g ` G )
49		mplcoe2.m	. . . . . 6 |- .^ = (.g ` G )
50	46, 48, 49	mulg0 17546	. . . . 5 |- ( ( V ` X ) e. ( Base ` P ) -> ( 0 .^ ( V ` X ) ) = ( 1r ` P ) )
51	44, 50	syl 17	. . . 4 |- ( ph -> ( 0 .^ ( V ` X ) ) = ( 1r ` P ) )
52		mplcoe1.d	. . . . 5 |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
53		mplcoe1.z	. . . . 5 |- .0. = ( 0g ` R )
54		mplcoe1.o	. . . . 5 |- .1. = ( 1r ` R )
55	38, 52, 53, 54, 47, 41, 42	mpl1 19444	. . . 4 |- ( ph -> ( 1r ` P ) = ( y e. D |-> if ( y = ( I X. { 0 } ) , .1. , .0. ) ) )
56	51, 55	eqtr2d 2657	. . 3 |- ( ph -> ( y e. D |-> if ( y = ( I X. { 0 } ) , .1. , .0. ) ) = ( 0 .^ ( V ` X ) ) )
57		oveq1 6657	. . . . . 6 |- ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) = ( n .^ ( V ` X ) ) -> ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) ( .r ` P ) ( V ` X ) ) = ( ( n .^ ( V ` X ) ) ( .r ` P ) ( V ` X ) ) )
58	41	adantr 481	. . . . . . . . 9 |- ( ( ph /\ n e. NN0 ) -> I e. W )
59	42	adantr 481	. . . . . . . . 9 |- ( ( ph /\ n e. NN0 ) -> R e. Ring )
60	52	snifpsrbag 19366	. . . . . . . . . 10 |- ( ( I e. W /\ n e. NN0 ) -> ( k e. I |-> if ( k = X , n , 0 ) ) e. D )
61	41, 60	sylan 488	. . . . . . . . 9 |- ( ( ph /\ n e. NN0 ) -> ( k e. I |-> if ( k = X , n , 0 ) ) e. D )
62		eqid 2622	. . . . . . . . 9 |- ( .r ` P ) = ( .r ` P )
63		1nn0 11308	. . . . . . . . . . 11 |- 1 e. NN0
64	63	a1i 11	. . . . . . . . . 10 |- ( n e. NN0 -> 1 e. NN0 )
65	52	snifpsrbag 19366	. . . . . . . . . 10 |- ( ( I e. W /\ 1 e. NN0 ) -> ( k e. I |-> if ( k = X , 1 , 0 ) ) e. D )
66	41, 64, 65	syl2an 494	. . . . . . . . 9 |- ( ( ph /\ n e. NN0 ) -> ( k e. I |-> if ( k = X , 1 , 0 ) ) e. D )
67	38, 40, 53, 54, 52, 58, 59, 61, 62, 66	mplmonmul 19464	. . . . . . . 8 |- ( ( ph /\ n e. NN0 ) -> ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) ( .r ` P ) ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , 1 , 0 ) ) , .1. , .0. ) ) ) = ( y e. D |-> if ( y = ( ( k e. I |-> if ( k = X , n , 0 ) ) oF + ( k e. I |-> if ( k = X , 1 , 0 ) ) ) , .1. , .0. ) ) )
68	43	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN0 ) -> X e. I )
69	39, 52, 53, 54, 58, 59, 68	mvrval 19421	. . . . . . . . . 10 |- ( ( ph /\ n e. NN0 ) -> ( V ` X ) = ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , 1 , 0 ) ) , .1. , .0. ) ) )
70	69	eqcomd 2628	. . . . . . . . 9 |- ( ( ph /\ n e. NN0 ) -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , 1 , 0 ) ) , .1. , .0. ) ) = ( V ` X ) )
71	70	oveq2d 6666	. . . . . . . 8 |- ( ( ph /\ n e. NN0 ) -> ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) ( .r ` P ) ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , 1 , 0 ) ) , .1. , .0. ) ) ) = ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) ( .r ` P ) ( V ` X ) ) )
72		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. NN0 ) /\ k e. I ) -> n e. NN0 )
73		0nn0 11307	. . . . . . . . . . . . . 14 |- 0 e. NN0
74		ifcl 4130	. . . . . . . . . . . . . 14 |- ( ( n e. NN0 /\ 0 e. NN0 ) -> if ( k = X , n , 0 ) e. NN0 )
75	72, 73, 74	sylancl 694	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. NN0 ) /\ k e. I ) -> if ( k = X , n , 0 ) e. NN0 )
76	63, 73	keepel 4155	. . . . . . . . . . . . . 14 |- if ( k = X , 1 , 0 ) e. NN0
77	76	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. NN0 ) /\ k e. I ) -> if ( k = X , 1 , 0 ) e. NN0 )
78		eqidd 2623	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN0 ) -> ( k e. I |-> if ( k = X , n , 0 ) ) = ( k e. I |-> if ( k = X , n , 0 ) ) )
79		eqidd 2623	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN0 ) -> ( k e. I |-> if ( k = X , 1 , 0 ) ) = ( k e. I |-> if ( k = X , 1 , 0 ) ) )
80	58, 75, 77, 78, 79	offval2 6914	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN0 ) -> ( ( k e. I |-> if ( k = X , n , 0 ) ) oF + ( k e. I |-> if ( k = X , 1 , 0 ) ) ) = ( k e. I |-> ( if ( k = X , n , 0 ) + if ( k = X , 1 , 0 ) ) ) )
81		iftrue 4092	. . . . . . . . . . . . . . . 16 |- ( k = X -> if ( k = X , n , 0 ) = n )
82		iftrue 4092	. . . . . . . . . . . . . . . 16 |- ( k = X -> if ( k = X , 1 , 0 ) = 1 )
83	81, 82	oveq12d 6668	. . . . . . . . . . . . . . 15 |- ( k = X -> ( if ( k = X , n , 0 ) + if ( k = X , 1 , 0 ) ) = ( n + 1 ) )
84		iftrue 4092	. . . . . . . . . . . . . . 15 |- ( k = X -> if ( k = X , ( n + 1 ) , 0 ) = ( n + 1 ) )
85	83, 84	eqtr4d 2659	. . . . . . . . . . . . . 14 |- ( k = X -> ( if ( k = X , n , 0 ) + if ( k = X , 1 , 0 ) ) = if ( k = X , ( n + 1 ) , 0 ) )
86		00id 10211	. . . . . . . . . . . . . . 15 |- ( 0 + 0 ) = 0
87		iffalse 4095	. . . . . . . . . . . . . . . 16 |- ( -. k = X -> if ( k = X , n , 0 ) = 0 )
88		iffalse 4095	. . . . . . . . . . . . . . . 16 |- ( -. k = X -> if ( k = X , 1 , 0 ) = 0 )
89	87, 88	oveq12d 6668	. . . . . . . . . . . . . . 15 |- ( -. k = X -> ( if ( k = X , n , 0 ) + if ( k = X , 1 , 0 ) ) = ( 0 + 0 ) )
90		iffalse 4095	. . . . . . . . . . . . . . 15 |- ( -. k = X -> if ( k = X , ( n + 1 ) , 0 ) = 0 )
91	86, 89, 90	3eqtr4a 2682	. . . . . . . . . . . . . 14 |- ( -. k = X -> ( if ( k = X , n , 0 ) + if ( k = X , 1 , 0 ) ) = if ( k = X , ( n + 1 ) , 0 ) )
92	85, 91	pm2.61i 176	. . . . . . . . . . . . 13 |- ( if ( k = X , n , 0 ) + if ( k = X , 1 , 0 ) ) = if ( k = X , ( n + 1 ) , 0 )
93	92	mpteq2i 4741	. . . . . . . . . . . 12 |- ( k e. I |-> ( if ( k = X , n , 0 ) + if ( k = X , 1 , 0 ) ) ) = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) )
94	80, 93	syl6eq 2672	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN0 ) -> ( ( k e. I |-> if ( k = X , n , 0 ) ) oF + ( k e. I |-> if ( k = X , 1 , 0 ) ) ) = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) )
95	94	eqeq2d 2632	. . . . . . . . . 10 |- ( ( ph /\ n e. NN0 ) -> ( y = ( ( k e. I |-> if ( k = X , n , 0 ) ) oF + ( k e. I |-> if ( k = X , 1 , 0 ) ) ) <-> y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) ) )
96	95	ifbid 4108	. . . . . . . . 9 |- ( ( ph /\ n e. NN0 ) -> if ( y = ( ( k e. I |-> if ( k = X , n , 0 ) ) oF + ( k e. I |-> if ( k = X , 1 , 0 ) ) ) , .1. , .0. ) = if ( y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) , .1. , .0. ) )
97	96	mpteq2dv 4745	. . . . . . . 8 |- ( ( ph /\ n e. NN0 ) -> ( y e. D |-> if ( y = ( ( k e. I |-> if ( k = X , n , 0 ) ) oF + ( k e. I |-> if ( k = X , 1 , 0 ) ) ) , .1. , .0. ) ) = ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) , .1. , .0. ) ) )
98	67, 71, 97	3eqtr3rd 2665	. . . . . . 7 |- ( ( ph /\ n e. NN0 ) -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) , .1. , .0. ) ) = ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) ( .r ` P ) ( V ` X ) ) )
99	38	mplring 19452	. . . . . . . . . . 11 |- ( ( I e. W /\ R e. Ring ) -> P e. Ring )
100	41, 42, 99	syl2anc 693	. . . . . . . . . 10 |- ( ph -> P e. Ring )
101	45	ringmgp 18553	. . . . . . . . . 10 |- ( P e. Ring -> G e. Mnd )
102	100, 101	syl 17	. . . . . . . . 9 |- ( ph -> G e. Mnd )
103	102	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. NN0 ) -> G e. Mnd )
104		simpr 477	. . . . . . . 8 |- ( ( ph /\ n e. NN0 ) -> n e. NN0 )
105	44	adantr 481	. . . . . . . 8 |- ( ( ph /\ n e. NN0 ) -> ( V ` X ) e. ( Base ` P ) )
106	45, 62	mgpplusg 18493	. . . . . . . . 9 |- ( .r ` P ) = ( +g ` G )
107	46, 49, 106	mulgnn0p1 17552	. . . . . . . 8 |- ( ( G e. Mnd /\ n e. NN0 /\ ( V ` X ) e. ( Base ` P ) ) -> ( ( n + 1 ) .^ ( V ` X ) ) = ( ( n .^ ( V ` X ) ) ( .r ` P ) ( V ` X ) ) )
108	103, 104, 105, 107	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ n e. NN0 ) -> ( ( n + 1 ) .^ ( V ` X ) ) = ( ( n .^ ( V ` X ) ) ( .r ` P ) ( V ` X ) ) )
109	98, 108	eqeq12d 2637	. . . . . 6 |- ( ( ph /\ n e. NN0 ) -> ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) , .1. , .0. ) ) = ( ( n + 1 ) .^ ( V ` X ) ) <-> ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) ( .r ` P ) ( V ` X ) ) = ( ( n .^ ( V ` X ) ) ( .r ` P ) ( V ` X ) ) ) )
110	57, 109	syl5ibr 236	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) = ( n .^ ( V ` X ) ) -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) , .1. , .0. ) ) = ( ( n + 1 ) .^ ( V ` X ) ) ) )
111	110	expcom 451	. . . 4 |- ( n e. NN0 -> ( ph -> ( ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) = ( n .^ ( V ` X ) ) -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) , .1. , .0. ) ) = ( ( n + 1 ) .^ ( V ` X ) ) ) ) )
112	111	a2d 29	. . 3 |- ( n e. NN0 -> ( ( ph -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , n , 0 ) ) , .1. , .0. ) ) = ( n .^ ( V ` X ) ) ) -> ( ph -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , ( n + 1 ) , 0 ) ) , .1. , .0. ) ) = ( ( n + 1 ) .^ ( V ` X ) ) ) ) )
113	13, 21, 29, 37, 56, 112	nn0ind 11472	. 2 |- ( N e. NN0 -> ( ph -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , N , 0 ) ) , .1. , .0. ) ) = ( N .^ ( V ` X ) ) ) )
114	1, 113	mpcom 38	1 |- ( ph -> ( y e. D |-> if ( y = ( k e. I |-> if ( k = X , N , 0 ) ) , .1. , .0. ) ) = ( N .^ ( V ` X ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   ifcif 4086   {csn 4177   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   oFcof 6895   ^m cmap 7857   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   NNcn 11020   NN0cn0 11292   Basecbs 15857   .rcmulr 15942   0gc0g 16100   Mndcmnd 17294  ._gcmg 17540  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547   mVar cmvr 19352   mPoly cmpl 19353

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

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-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-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-sca 15957  df-vsca 15958  df-tset 15960  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-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-psr 19356  df-mvr 19357  df-mpl 19358

This theorem is referenced by:  mplcoe5  19468  coe1tm  19643