Theorem breprexpnat 30712

Description: Express the S th power of the finite series in terms of the number of representations of integers m as sums of S terms of elements of A, bounded by N. Proposition of [Nathanson] p. 123. (Contributed by Thierry Arnoux, 11-Dec-2021.)

Hypotheses

Ref	Expression
breprexp.n	|- ( ph -> N e. NN0 )
breprexp.s	|- ( ph -> S e. NN0 )
breprexp.z	|- ( ph -> Z e. CC )
breprexpnat.a	|- ( ph -> A C_ NN )
breprexpnat.p	|- P = sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b )
breprexpnat.r	|- R = ( # ` ( ( A i^i ( 1 ... N ) ) (repr ` S ) m ) )

Assertion

Ref	Expression
breprexpnat	|- ( ph -> ( P ^ S ) = sum_ m e. ( 0 ... ( S x. N ) ) ( R x. ( Z ^ m ) ) )

Distinct variable groups:   m, N   S, m   m, Z   A, b, m   N, b   S, b   Z, b   ph, b, m

Allowed substitution hints:   P( m, b)   R( m, b)

Proof of Theorem breprexpnat

Dummy variables c a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breprexp.n	. . . 4 |- ( ph -> N e. NN0 )
2		breprexp.s	. . . 4 |- ( ph -> S e. NN0 )
3		breprexp.z	. . . 4 |- ( ph -> Z e. CC )
4		fvex 6201	. . . . . 6 |- ( (𝟭 ` NN ) ` A ) e. _V
5	4	fconst 6091	. . . . 5 |- ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) : ( 0..^ S ) --> { ( (𝟭 ` NN ) ` A ) }
6		nnex 11026	. . . . . . . . 9 |- NN e. _V
7		breprexpnat.a	. . . . . . . . 9 |- ( ph -> A C_ NN )
8		indf 30077	. . . . . . . . 9 |- ( ( NN e. _V /\ A C_ NN ) -> ( (𝟭 ` NN ) ` A ) : NN --> { 0 , 1 } )
9	6, 7, 8	sylancr 695	. . . . . . . 8 |- ( ph -> ( (𝟭 ` NN ) ` A ) : NN --> { 0 , 1 } )
10		0cn 10032	. . . . . . . . 9 |- 0 e. CC
11		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
12		prssi 4353	. . . . . . . . 9 |- ( ( 0 e. CC /\ 1 e. CC ) -> { 0 , 1 } C_ CC )
13	10, 11, 12	mp2an 708	. . . . . . . 8 |- { 0 , 1 } C_ CC
14		fss 6056	. . . . . . . 8 |- ( ( ( (𝟭 ` NN ) ` A ) : NN --> { 0 , 1 } /\ { 0 , 1 } C_ CC ) -> ( (𝟭 ` NN ) ` A ) : NN --> CC )
15	9, 13, 14	sylancl 694	. . . . . . 7 |- ( ph -> ( (𝟭 ` NN ) ` A ) : NN --> CC )
16		cnex 10017	. . . . . . . 8 |- CC e. _V
17	16, 6	elmap 7886	. . . . . . 7 |- ( ( (𝟭 ` NN ) ` A ) e. ( CC ^m NN ) <-> ( (𝟭 ` NN ) ` A ) : NN --> CC )
18	15, 17	sylibr 224	. . . . . 6 |- ( ph -> ( (𝟭 ` NN ) ` A ) e. ( CC ^m NN ) )
19	4	snss 4316	. . . . . 6 |- ( ( (𝟭 ` NN ) ` A ) e. ( CC ^m NN ) <-> { ( (𝟭 ` NN ) ` A ) } C_ ( CC ^m NN ) )
20	18, 19	sylib 208	. . . . 5 |- ( ph -> { ( (𝟭 ` NN ) ` A ) } C_ ( CC ^m NN ) )
21		fss 6056	. . . . 5 |- ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) : ( 0..^ S ) --> { ( (𝟭 ` NN ) ` A ) } /\ { ( (𝟭 ` NN ) ` A ) } C_ ( CC ^m NN ) ) -> ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) : ( 0..^ S ) --> ( CC ^m NN ) )
22	5, 20, 21	sylancr 695	. . . 4 |- ( ph -> ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) : ( 0..^ S ) --> ( CC ^m NN ) )
23	1, 2, 3, 22	breprexp 30711	. . 3 |- ( ph -> prod_ a e. ( 0..^ S ) sum_ b e. ( 1 ... N ) ( ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` b ) x. ( Z ^ b ) ) = sum_ m e. ( 0 ... ( S x. N ) ) sum_ c e. ( ( 1 ... N ) (repr ` S ) m ) ( prod_ a e. ( 0..^ S ) ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` ( c ` a ) ) x. ( Z ^ m ) ) )
24	4	fvconst2 6469	. . . . . . . . . 10 |- ( a e. ( 0..^ S ) -> ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) = ( (𝟭 ` NN ) ` A ) )
25	24	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ph /\ a e. ( 0..^ S ) ) /\ b e. ( 1 ... N ) ) -> ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) = ( (𝟭 ` NN ) ` A ) )
26	25	fveq1d 6193	. . . . . . . 8 |- ( ( ( ph /\ a e. ( 0..^ S ) ) /\ b e. ( 1 ... N ) ) -> ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` b ) = ( ( (𝟭 ` NN ) ` A ) ` b ) )
27	26	oveq1d 6665	. . . . . . 7 |- ( ( ( ph /\ a e. ( 0..^ S ) ) /\ b e. ( 1 ... N ) ) -> ( ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` b ) x. ( Z ^ b ) ) = ( ( ( (𝟭 ` NN ) ` A ) ` b ) x. ( Z ^ b ) ) )
28	27	sumeq2dv 14433	. . . . . 6 |- ( ( ph /\ a e. ( 0..^ S ) ) -> sum_ b e. ( 1 ... N ) ( ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` b ) x. ( Z ^ b ) ) = sum_ b e. ( 1 ... N ) ( ( ( (𝟭 ` NN ) ` A ) ` b ) x. ( Z ^ b ) ) )
29	6	a1i 11	. . . . . . 7 |- ( ( ph /\ a e. ( 0..^ S ) ) -> NN e. _V )
30		fzfi 12771	. . . . . . . 8 |- ( 1 ... N ) e. Fin
31	30	a1i 11	. . . . . . 7 |- ( ( ph /\ a e. ( 0..^ S ) ) -> ( 1 ... N ) e. Fin )
32		fz1ssnn 12372	. . . . . . . 8 |- ( 1 ... N ) C_ NN
33	32	a1i 11	. . . . . . 7 |- ( ( ph /\ a e. ( 0..^ S ) ) -> ( 1 ... N ) C_ NN )
34	7	adantr 481	. . . . . . 7 |- ( ( ph /\ a e. ( 0..^ S ) ) -> A C_ NN )
35	3	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ a e. ( 0..^ S ) ) /\ b e. ( 1 ... N ) ) -> Z e. CC )
36		nnssnn0 11295	. . . . . . . . . 10 |- NN C_ NN0
37	32, 36	sstri 3612	. . . . . . . . 9 |- ( 1 ... N ) C_ NN0
38		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ a e. ( 0..^ S ) ) /\ b e. ( 1 ... N ) ) -> b e. ( 1 ... N ) )
39	37, 38	sseldi 3601	. . . . . . . 8 |- ( ( ( ph /\ a e. ( 0..^ S ) ) /\ b e. ( 1 ... N ) ) -> b e. NN0 )
40	35, 39	expcld 13008	. . . . . . 7 |- ( ( ( ph /\ a e. ( 0..^ S ) ) /\ b e. ( 1 ... N ) ) -> ( Z ^ b ) e. CC )
41	29, 31, 33, 34, 40	indsumin 30084	. . . . . 6 |- ( ( ph /\ a e. ( 0..^ S ) ) -> sum_ b e. ( 1 ... N ) ( ( ( (𝟭 ` NN ) ` A ) ` b ) x. ( Z ^ b ) ) = sum_ b e. ( ( 1 ... N ) i^i A ) ( Z ^ b ) )
42		incom 3805	. . . . . . . 8 |- ( ( 1 ... N ) i^i A ) = ( A i^i ( 1 ... N ) )
43	42	a1i 11	. . . . . . 7 |- ( ( ph /\ a e. ( 0..^ S ) ) -> ( ( 1 ... N ) i^i A ) = ( A i^i ( 1 ... N ) ) )
44	43	sumeq1d 14431	. . . . . 6 |- ( ( ph /\ a e. ( 0..^ S ) ) -> sum_ b e. ( ( 1 ... N ) i^i A ) ( Z ^ b ) = sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) )
45	28, 41, 44	3eqtrd 2660	. . . . 5 |- ( ( ph /\ a e. ( 0..^ S ) ) -> sum_ b e. ( 1 ... N ) ( ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` b ) x. ( Z ^ b ) ) = sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) )
46	45	prodeq2dv 14653	. . . 4 |- ( ph -> prod_ a e. ( 0..^ S ) sum_ b e. ( 1 ... N ) ( ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` b ) x. ( Z ^ b ) ) = prod_ a e. ( 0..^ S ) sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) )
47		fzofi 12773	. . . . . 6 |- ( 0..^ S ) e. Fin
48	47	a1i 11	. . . . 5 |- ( ph -> ( 0..^ S ) e. Fin )
49		inss2 3834	. . . . . . . 8 |- ( A i^i ( 1 ... N ) ) C_ ( 1 ... N )
50		ssfi 8180	. . . . . . . 8 |- ( ( ( 1 ... N ) e. Fin /\ ( A i^i ( 1 ... N ) ) C_ ( 1 ... N ) ) -> ( A i^i ( 1 ... N ) ) e. Fin )
51	30, 49, 50	mp2an 708	. . . . . . 7 |- ( A i^i ( 1 ... N ) ) e. Fin
52	51	a1i 11	. . . . . 6 |- ( ph -> ( A i^i ( 1 ... N ) ) e. Fin )
53	3	adantr 481	. . . . . . 7 |- ( ( ph /\ b e. ( A i^i ( 1 ... N ) ) ) -> Z e. CC )
54	49, 37	sstri 3612	. . . . . . . 8 |- ( A i^i ( 1 ... N ) ) C_ NN0
55		simpr 477	. . . . . . . 8 |- ( ( ph /\ b e. ( A i^i ( 1 ... N ) ) ) -> b e. ( A i^i ( 1 ... N ) ) )
56	54, 55	sseldi 3601	. . . . . . 7 |- ( ( ph /\ b e. ( A i^i ( 1 ... N ) ) ) -> b e. NN0 )
57	53, 56	expcld 13008	. . . . . 6 |- ( ( ph /\ b e. ( A i^i ( 1 ... N ) ) ) -> ( Z ^ b ) e. CC )
58	52, 57	fsumcl 14464	. . . . 5 |- ( ph -> sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) e. CC )
59		fprodconst 14708	. . . . 5 |- ( ( ( 0..^ S ) e. Fin /\ sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) e. CC ) -> prod_ a e. ( 0..^ S ) sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) = ( sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) ^ ( # ` ( 0..^ S ) ) ) )
60	48, 58, 59	syl2anc 693	. . . 4 |- ( ph -> prod_ a e. ( 0..^ S ) sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) = ( sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) ^ ( # ` ( 0..^ S ) ) ) )
61		hashfzo0 13217	. . . . . 6 |- ( S e. NN0 -> ( # ` ( 0..^ S ) ) = S )
62	2, 61	syl 17	. . . . 5 |- ( ph -> ( # ` ( 0..^ S ) ) = S )
63	62	oveq2d 6666	. . . 4 |- ( ph -> ( sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) ^ ( # ` ( 0..^ S ) ) ) = ( sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) ^ S ) )
64	46, 60, 63	3eqtrd 2660	. . 3 |- ( ph -> prod_ a e. ( 0..^ S ) sum_ b e. ( 1 ... N ) ( ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` b ) x. ( Z ^ b ) ) = ( sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) ^ S ) )
65	32	a1i 11	. . . . . . 7 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> ( 1 ... N ) C_ NN )
66		fzssz 12343	. . . . . . . 8 |- ( 0 ... ( S x. N ) ) C_ ZZ
67		simpr 477	. . . . . . . 8 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> m e. ( 0 ... ( S x. N ) ) )
68	66, 67	sseldi 3601	. . . . . . 7 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> m e. ZZ )
69	2	adantr 481	. . . . . . 7 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> S e. NN0 )
70	30	a1i 11	. . . . . . 7 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> ( 1 ... N ) e. Fin )
71	65, 68, 69, 70	reprfi 30694	. . . . . 6 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> ( ( 1 ... N ) (repr ` S ) m ) e. Fin )
72	3	adantr 481	. . . . . . 7 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> Z e. CC )
73		fz0ssnn0 12435	. . . . . . . 8 |- ( 0 ... ( S x. N ) ) C_ NN0
74	73, 67	sseldi 3601	. . . . . . 7 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> m e. NN0 )
75	72, 74	expcld 13008	. . . . . 6 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> ( Z ^ m ) e. CC )
76	47	a1i 11	. . . . . . 7 |- ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) -> ( 0..^ S ) e. Fin )
77	9	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) /\ a e. ( 0..^ S ) ) -> ( (𝟭 ` NN ) ` A ) : NN --> { 0 , 1 } )
78	32	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) -> ( 1 ... N ) C_ NN )
79	68	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) -> m e. ZZ )
80	69	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) -> S e. NN0 )
81		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) -> c e. ( ( 1 ... N ) (repr ` S ) m ) )
82	78, 79, 80, 81	reprf 30690	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) -> c : ( 0..^ S ) --> ( 1 ... N ) )
83	82	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) /\ a e. ( 0..^ S ) ) -> ( c ` a ) e. ( 1 ... N ) )
84	32, 83	sseldi 3601	. . . . . . . . 9 |- ( ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) /\ a e. ( 0..^ S ) ) -> ( c ` a ) e. NN )
85	77, 84	ffvelrnd 6360	. . . . . . . 8 |- ( ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) /\ a e. ( 0..^ S ) ) -> ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) e. { 0 , 1 } )
86	13, 85	sseldi 3601	. . . . . . 7 |- ( ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) /\ a e. ( 0..^ S ) ) -> ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) e. CC )
87	76, 86	fprodcl 14682	. . . . . 6 |- ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) -> prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) e. CC )
88	71, 75, 87	fsummulc1 14517	. . . . 5 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> ( sum_ c e. ( ( 1 ... N ) (repr ` S ) m ) prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) x. ( Z ^ m ) ) = sum_ c e. ( ( 1 ... N ) (repr ` S ) m ) ( prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) x. ( Z ^ m ) ) )
89	7	adantr 481	. . . . . . 7 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> A C_ NN )
90	89, 68, 69, 70, 65	hashreprin 30698	. . . . . 6 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> ( # ` ( ( A i^i ( 1 ... N ) ) (repr ` S ) m ) ) = sum_ c e. ( ( 1 ... N ) (repr ` S ) m ) prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) )
91	90	oveq1d 6665	. . . . 5 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> ( ( # ` ( ( A i^i ( 1 ... N ) ) (repr ` S ) m ) ) x. ( Z ^ m ) ) = ( sum_ c e. ( ( 1 ... N ) (repr ` S ) m ) prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) x. ( Z ^ m ) ) )
92	24	fveq1d 6193	. . . . . . . . . 10 |- ( a e. ( 0..^ S ) -> ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` ( c ` a ) ) = ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) )
93	92	adantl 482	. . . . . . . . 9 |- ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ a e. ( 0..^ S ) ) -> ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` ( c ` a ) ) = ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) )
94	93	prodeq2dv 14653	. . . . . . . 8 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> prod_ a e. ( 0..^ S ) ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` ( c ` a ) ) = prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) )
95	94	adantr 481	. . . . . . 7 |- ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) -> prod_ a e. ( 0..^ S ) ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` ( c ` a ) ) = prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) )
96	95	oveq1d 6665	. . . . . 6 |- ( ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) /\ c e. ( ( 1 ... N ) (repr ` S ) m ) ) -> ( prod_ a e. ( 0..^ S ) ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` ( c ` a ) ) x. ( Z ^ m ) ) = ( prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) x. ( Z ^ m ) ) )
97	96	sumeq2dv 14433	. . . . 5 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> sum_ c e. ( ( 1 ... N ) (repr ` S ) m ) ( prod_ a e. ( 0..^ S ) ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` ( c ` a ) ) x. ( Z ^ m ) ) = sum_ c e. ( ( 1 ... N ) (repr ` S ) m ) ( prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) x. ( Z ^ m ) ) )
98	88, 91, 97	3eqtr4rd 2667	. . . 4 |- ( ( ph /\ m e. ( 0 ... ( S x. N ) ) ) -> sum_ c e. ( ( 1 ... N ) (repr ` S ) m ) ( prod_ a e. ( 0..^ S ) ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` ( c ` a ) ) x. ( Z ^ m ) ) = ( ( # ` ( ( A i^i ( 1 ... N ) ) (repr ` S ) m ) ) x. ( Z ^ m ) ) )
99	98	sumeq2dv 14433	. . 3 |- ( ph -> sum_ m e. ( 0 ... ( S x. N ) ) sum_ c e. ( ( 1 ... N ) (repr ` S ) m ) ( prod_ a e. ( 0..^ S ) ( ( ( ( 0..^ S ) X. { ( (𝟭 ` NN ) ` A ) } ) ` a ) ` ( c ` a ) ) x. ( Z ^ m ) ) = sum_ m e. ( 0 ... ( S x. N ) ) ( ( # ` ( ( A i^i ( 1 ... N ) ) (repr ` S ) m ) ) x. ( Z ^ m ) ) )
100	23, 64, 99	3eqtr3d 2664	. 2 |- ( ph -> ( sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) ^ S ) = sum_ m e. ( 0 ... ( S x. N ) ) ( ( # ` ( ( A i^i ( 1 ... N ) ) (repr ` S ) m ) ) x. ( Z ^ m ) ) )
101		breprexpnat.p	. . 3 |- P = sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b )
102	101	oveq1i 6660	. 2 |- ( P ^ S ) = ( sum_ b e. ( A i^i ( 1 ... N ) ) ( Z ^ b ) ^ S )
103		breprexpnat.r	. . . . 5 |- R = ( # ` ( ( A i^i ( 1 ... N ) ) (repr ` S ) m ) )
104	103	oveq1i 6660	. . . 4 |- ( R x. ( Z ^ m ) ) = ( ( # ` ( ( A i^i ( 1 ... N ) ) (repr ` S ) m ) ) x. ( Z ^ m ) )
105	104	a1i 11	. . 3 |- ( m e. ( 0 ... ( S x. N ) ) -> ( R x. ( Z ^ m ) ) = ( ( # ` ( ( A i^i ( 1 ... N ) ) (repr ` S ) m ) ) x. ( Z ^ m ) ) )
106	105	sumeq2i 14429	. 2 |- sum_ m e. ( 0 ... ( S x. N ) ) ( R x. ( Z ^ m ) ) = sum_ m e. ( 0 ... ( S x. N ) ) ( ( # ` ( ( A i^i ( 1 ... N ) ) (repr ` S ) m ) ) x. ( Z ^ m ) )
107	100, 102, 106	3eqtr4g 2681	1 |- ( ph -> ( P ^ S ) = sum_ m e. ( 0 ... ( S x. N ) ) ( R x. ( Z ^ m ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   {cpr 4179   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326  ..^cfzo 12465   ^cexp 12860   #chash 13117   sum_csu 14416   prod_cprod 14635  𝟭cind 30072  reprcrepr 30686

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

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-disj 4621  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-om 7066  df-1st 7168  df-2nd 7169  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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  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-sum 14417  df-prod 14636  df-ind 30073  df-repr 30687

This theorem is referenced by: (None)