Theorem fsumcom2 14505

Description: Interchange order of summation. Note that B ( j ) and D ( k ) are not necessarily constant expressions. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.) (Proof shortened by JJ, 2-Aug-2021.)

Hypotheses

Ref	Expression
fsumcom2.1	|- ( ph -> A e. Fin )
fsumcom2.2	|- ( ph -> C e. Fin )
fsumcom2.3	|- ( ( ph /\ j e. A ) -> B e. Fin )
fsumcom2.4	|- ( ph -> ( ( j e. A /\ k e. B ) <-> ( k e. C /\ j e. D ) ) )
fsumcom2.5	|- ( ( ph /\ ( j e. A /\ k e. B ) ) -> E e. CC )

Assertion

Ref	Expression
fsumcom2	|- ( ph -> sum_ j e. A sum_ k e. B E = sum_ k e. C sum_ j e. D E )

Distinct variable groups:   j, k, A   C, j, k   ph, j, k   B, k   D, j

Allowed substitution hints:   B( j)   D( k)   E( j, k)

Proof of Theorem fsumcom2

Dummy variables m n x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 5227	. . . . . . . . 9 |- Rel ( { j } X. B )
2	1	rgenw 2924	. . . . . . . 8 |- A. j e. A Rel ( { j } X. B )
3		reliun 5239	. . . . . . . 8 |- ( Rel U_ j e. A ( { j } X. B ) <-> A. j e. A Rel ( { j } X. B ) )
4	2, 3	mpbir 221	. . . . . . 7 |- Rel U_ j e. A ( { j } X. B )
5		relcnv 5503	. . . . . . 7 |- Rel `' U_ k e. C ( { k } X. D )
6		ancom 466	. . . . . . . . . . . 12 |- ( ( x = j /\ y = k ) <-> ( y = k /\ x = j ) )
7		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
8		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
9	7, 8	opth 4945	. . . . . . . . . . . 12 |- ( <. x , y >. = <. j , k >. <-> ( x = j /\ y = k ) )
10	8, 7	opth 4945	. . . . . . . . . . . 12 |- ( <. y , x >. = <. k , j >. <-> ( y = k /\ x = j ) )
11	6, 9, 10	3bitr4i 292	. . . . . . . . . . 11 |- ( <. x , y >. = <. j , k >. <-> <. y , x >. = <. k , j >. )
12	11	a1i 11	. . . . . . . . . 10 |- ( ph -> ( <. x , y >. = <. j , k >. <-> <. y , x >. = <. k , j >. ) )
13		fsumcom2.4	. . . . . . . . . 10 |- ( ph -> ( ( j e. A /\ k e. B ) <-> ( k e. C /\ j e. D ) ) )
14	12, 13	anbi12d 747	. . . . . . . . 9 |- ( ph -> ( ( <. x , y >. = <. j , k >. /\ ( j e. A /\ k e. B ) ) <-> ( <. y , x >. = <. k , j >. /\ ( k e. C /\ j e. D ) ) ) )
15	14	2exbidv 1852	. . . . . . . 8 |- ( ph -> ( E. j E. k ( <. x , y >. = <. j , k >. /\ ( j e. A /\ k e. B ) ) <-> E. j E. k ( <. y , x >. = <. k , j >. /\ ( k e. C /\ j e. D ) ) ) )
16		eliunxp 5259	. . . . . . . 8 |- ( <. x , y >. e. U_ j e. A ( { j } X. B ) <-> E. j E. k ( <. x , y >. = <. j , k >. /\ ( j e. A /\ k e. B ) ) )
17	7, 8	opelcnv 5304	. . . . . . . . 9 |- ( <. x , y >. e. `' U_ k e. C ( { k } X. D ) <-> <. y , x >. e. U_ k e. C ( { k } X. D ) )
18		eliunxp 5259	. . . . . . . . 9 |- ( <. y , x >. e. U_ k e. C ( { k } X. D ) <-> E. k E. j ( <. y , x >. = <. k , j >. /\ ( k e. C /\ j e. D ) ) )
19		excom 2042	. . . . . . . . 9 |- ( E. k E. j ( <. y , x >. = <. k , j >. /\ ( k e. C /\ j e. D ) ) <-> E. j E. k ( <. y , x >. = <. k , j >. /\ ( k e. C /\ j e. D ) ) )
20	17, 18, 19	3bitri 286	. . . . . . . 8 |- ( <. x , y >. e. `' U_ k e. C ( { k } X. D ) <-> E. j E. k ( <. y , x >. = <. k , j >. /\ ( k e. C /\ j e. D ) ) )
21	15, 16, 20	3bitr4g 303	. . . . . . 7 |- ( ph -> ( <. x , y >. e. U_ j e. A ( { j } X. B ) <-> <. x , y >. e. `' U_ k e. C ( { k } X. D ) ) )
22	4, 5, 21	eqrelrdv 5216	. . . . . 6 |- ( ph -> U_ j e. A ( { j } X. B ) = `' U_ k e. C ( { k } X. D ) )
23		nfcv 2764	. . . . . . 7 |- F/_ m ( { j } X. B )
24		nfcv 2764	. . . . . . . 8 |- F/_ j { m }
25		nfcsb1v 3549	. . . . . . . 8 |- F/_ j [_ m / j ]_ B
26	24, 25	nfxp 5142	. . . . . . 7 |- F/_ j ( { m } X. [_ m / j ]_ B )
27		sneq 4187	. . . . . . . 8 |- ( j = m -> { j } = { m } )
28		csbeq1a 3542	. . . . . . . 8 |- ( j = m -> B = [_ m / j ]_ B )
29	27, 28	xpeq12d 5140	. . . . . . 7 |- ( j = m -> ( { j } X. B ) = ( { m } X. [_ m / j ]_ B ) )
30	23, 26, 29	cbviun 4557	. . . . . 6 |- U_ j e. A ( { j } X. B ) = U_ m e. A ( { m } X. [_ m / j ]_ B )
31		nfcv 2764	. . . . . . . 8 |- F/_ n ( { k } X. D )
32		nfcv 2764	. . . . . . . . 9 |- F/_ k { n }
33		nfcsb1v 3549	. . . . . . . . 9 |- F/_ k [_ n / k ]_ D
34	32, 33	nfxp 5142	. . . . . . . 8 |- F/_ k ( { n } X. [_ n / k ]_ D )
35		sneq 4187	. . . . . . . . 9 |- ( k = n -> { k } = { n } )
36		csbeq1a 3542	. . . . . . . . 9 |- ( k = n -> D = [_ n / k ]_ D )
37	35, 36	xpeq12d 5140	. . . . . . . 8 |- ( k = n -> ( { k } X. D ) = ( { n } X. [_ n / k ]_ D ) )
38	31, 34, 37	cbviun 4557	. . . . . . 7 |- U_ k e. C ( { k } X. D ) = U_ n e. C ( { n } X. [_ n / k ]_ D )
39	38	cnveqi 5297	. . . . . 6 |- `' U_ k e. C ( { k } X. D ) = `' U_ n e. C ( { n } X. [_ n / k ]_ D )
40	22, 30, 39	3eqtr3g 2679	. . . . 5 |- ( ph -> U_ m e. A ( { m } X. [_ m / j ]_ B ) = `' U_ n e. C ( { n } X. [_ n / k ]_ D ) )
41	40	sumeq1d 14431	. . . 4 |- ( ph -> sum_ z e. U_ m e. A ( { m } X. [_ m / j ]_ B ) [_ ( 2nd ` z ) / k ]_ [_ ( 1st ` z ) / j ]_ E = sum_ z e. `' U_ n e. C ( { n } X. [_ n / k ]_ D ) [_ ( 2nd ` z ) / k ]_ [_ ( 1st ` z ) / j ]_ E )
42		vex 3203	. . . . . . . 8 |- n e. _V
43		vex 3203	. . . . . . . 8 |- m e. _V
44	42, 43	op1std 7178	. . . . . . 7 |- ( w = <. n , m >. -> ( 1st ` w ) = n )
45	44	csbeq1d 3540	. . . . . 6 |- ( w = <. n , m >. -> [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E = [_ n / k ]_ [_ ( 2nd ` w ) / j ]_ E )
46	42, 43	op2ndd 7179	. . . . . . . 8 |- ( w = <. n , m >. -> ( 2nd ` w ) = m )
47	46	csbeq1d 3540	. . . . . . 7 |- ( w = <. n , m >. -> [_ ( 2nd ` w ) / j ]_ E = [_ m / j ]_ E )
48	47	csbeq2dv 3992	. . . . . 6 |- ( w = <. n , m >. -> [_ n / k ]_ [_ ( 2nd ` w ) / j ]_ E = [_ n / k ]_ [_ m / j ]_ E )
49	45, 48	eqtrd 2656	. . . . 5 |- ( w = <. n , m >. -> [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E = [_ n / k ]_ [_ m / j ]_ E )
50	43, 42	op2ndd 7179	. . . . . . 7 |- ( z = <. m , n >. -> ( 2nd ` z ) = n )
51	50	csbeq1d 3540	. . . . . 6 |- ( z = <. m , n >. -> [_ ( 2nd ` z ) / k ]_ [_ ( 1st ` z ) / j ]_ E = [_ n / k ]_ [_ ( 1st ` z ) / j ]_ E )
52	43, 42	op1std 7178	. . . . . . . 8 |- ( z = <. m , n >. -> ( 1st ` z ) = m )
53	52	csbeq1d 3540	. . . . . . 7 |- ( z = <. m , n >. -> [_ ( 1st ` z ) / j ]_ E = [_ m / j ]_ E )
54	53	csbeq2dv 3992	. . . . . 6 |- ( z = <. m , n >. -> [_ n / k ]_ [_ ( 1st ` z ) / j ]_ E = [_ n / k ]_ [_ m / j ]_ E )
55	51, 54	eqtrd 2656	. . . . 5 |- ( z = <. m , n >. -> [_ ( 2nd ` z ) / k ]_ [_ ( 1st ` z ) / j ]_ E = [_ n / k ]_ [_ m / j ]_ E )
56		fsumcom2.2	. . . . . 6 |- ( ph -> C e. Fin )
57		snfi 8038	. . . . . . . 8 |- { n } e. Fin
58		fsumcom2.1	. . . . . . . . . 10 |- ( ph -> A e. Fin )
59	58	adantr 481	. . . . . . . . 9 |- ( ( ph /\ n e. C ) -> A e. Fin )
60	43, 42	opelcnv 5304	. . . . . . . . . . . . . . . 16 |- ( <. m , n >. e. `' U_ k e. C ( { k } X. D ) <-> <. n , m >. e. U_ k e. C ( { k } X. D ) )
61	33, 36	opeliunxp2f 7336	. . . . . . . . . . . . . . . 16 |- ( <. n , m >. e. U_ k e. C ( { k } X. D ) <-> ( n e. C /\ m e. [_ n / k ]_ D ) )
62	60, 61	sylbbr 226	. . . . . . . . . . . . . . 15 |- ( ( n e. C /\ m e. [_ n / k ]_ D ) -> <. m , n >. e. `' U_ k e. C ( { k } X. D ) )
63	62	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> <. m , n >. e. `' U_ k e. C ( { k } X. D ) )
64	22	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> U_ j e. A ( { j } X. B ) = `' U_ k e. C ( { k } X. D ) )
65	63, 64	eleqtrrd 2704	. . . . . . . . . . . . 13 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> <. m , n >. e. U_ j e. A ( { j } X. B ) )
66		eliun 4524	. . . . . . . . . . . . 13 |- ( <. m , n >. e. U_ j e. A ( { j } X. B ) <-> E. j e. A <. m , n >. e. ( { j } X. B ) )
67	65, 66	sylib 208	. . . . . . . . . . . 12 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> E. j e. A <. m , n >. e. ( { j } X. B ) )
68		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( j e. A /\ <. m , n >. e. ( { j } X. B ) ) -> <. m , n >. e. ( { j } X. B ) )
69		opelxp 5146	. . . . . . . . . . . . . . . . 17 |- ( <. m , n >. e. ( { j } X. B ) <-> ( m e. { j } /\ n e. B ) )
70	68, 69	sylib 208	. . . . . . . . . . . . . . . 16 |- ( ( j e. A /\ <. m , n >. e. ( { j } X. B ) ) -> ( m e. { j } /\ n e. B ) )
71	70	simpld 475	. . . . . . . . . . . . . . 15 |- ( ( j e. A /\ <. m , n >. e. ( { j } X. B ) ) -> m e. { j } )
72		elsni 4194	. . . . . . . . . . . . . . 15 |- ( m e. { j } -> m = j )
73	71, 72	syl 17	. . . . . . . . . . . . . 14 |- ( ( j e. A /\ <. m , n >. e. ( { j } X. B ) ) -> m = j )
74		simpl 473	. . . . . . . . . . . . . 14 |- ( ( j e. A /\ <. m , n >. e. ( { j } X. B ) ) -> j e. A )
75	73, 74	eqeltrd 2701	. . . . . . . . . . . . 13 |- ( ( j e. A /\ <. m , n >. e. ( { j } X. B ) ) -> m e. A )
76	75	rexlimiva 3028	. . . . . . . . . . . 12 |- ( E. j e. A <. m , n >. e. ( { j } X. B ) -> m e. A )
77	67, 76	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> m e. A )
78	77	expr 643	. . . . . . . . . 10 |- ( ( ph /\ n e. C ) -> ( m e. [_ n / k ]_ D -> m e. A ) )
79	78	ssrdv 3609	. . . . . . . . 9 |- ( ( ph /\ n e. C ) -> [_ n / k ]_ D C_ A )
80	59, 79	ssfid 8183	. . . . . . . 8 |- ( ( ph /\ n e. C ) -> [_ n / k ]_ D e. Fin )
81		xpfi 8231	. . . . . . . 8 |- ( ( { n } e. Fin /\ [_ n / k ]_ D e. Fin ) -> ( { n } X. [_ n / k ]_ D ) e. Fin )
82	57, 80, 81	sylancr 695	. . . . . . 7 |- ( ( ph /\ n e. C ) -> ( { n } X. [_ n / k ]_ D ) e. Fin )
83	82	ralrimiva 2966	. . . . . 6 |- ( ph -> A. n e. C ( { n } X. [_ n / k ]_ D ) e. Fin )
84		iunfi 8254	. . . . . 6 |- ( ( C e. Fin /\ A. n e. C ( { n } X. [_ n / k ]_ D ) e. Fin ) -> U_ n e. C ( { n } X. [_ n / k ]_ D ) e. Fin )
85	56, 83, 84	syl2anc 693	. . . . 5 |- ( ph -> U_ n e. C ( { n } X. [_ n / k ]_ D ) e. Fin )
86		reliun 5239	. . . . . . 7 |- ( Rel U_ n e. C ( { n } X. [_ n / k ]_ D ) <-> A. n e. C Rel ( { n } X. [_ n / k ]_ D ) )
87		relxp 5227	. . . . . . . 8 |- Rel ( { n } X. [_ n / k ]_ D )
88	87	a1i 11	. . . . . . 7 |- ( n e. C -> Rel ( { n } X. [_ n / k ]_ D ) )
89	86, 88	mprgbir 2927	. . . . . 6 |- Rel U_ n e. C ( { n } X. [_ n / k ]_ D )
90	89	a1i 11	. . . . 5 |- ( ph -> Rel U_ n e. C ( { n } X. [_ n / k ]_ D ) )
91		simpr 477	. . . . . . . 8 |- ( ( ph /\ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) ) -> w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) )
92		eliun 4524	. . . . . . . 8 |- ( w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) <-> E. n e. C w e. ( { n } X. [_ n / k ]_ D ) )
93	91, 92	sylib 208	. . . . . . 7 |- ( ( ph /\ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) ) -> E. n e. C w e. ( { n } X. [_ n / k ]_ D ) )
94		xp2nd 7199	. . . . . . . . . 10 |- ( w e. ( { n } X. [_ n / k ]_ D ) -> ( 2nd ` w ) e. [_ n / k ]_ D )
95	94	adantl 482	. . . . . . . . 9 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> ( 2nd ` w ) e. [_ n / k ]_ D )
96		xp1st 7198	. . . . . . . . . . . 12 |- ( w e. ( { n } X. [_ n / k ]_ D ) -> ( 1st ` w ) e. { n } )
97	96	adantl 482	. . . . . . . . . . 11 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> ( 1st ` w ) e. { n } )
98		elsni 4194	. . . . . . . . . . 11 |- ( ( 1st ` w ) e. { n } -> ( 1st ` w ) = n )
99	97, 98	syl 17	. . . . . . . . . 10 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> ( 1st ` w ) = n )
100	99	csbeq1d 3540	. . . . . . . . 9 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> [_ ( 1st ` w ) / k ]_ D = [_ n / k ]_ D )
101	95, 100	eleqtrrd 2704	. . . . . . . 8 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> ( 2nd ` w ) e. [_ ( 1st ` w ) / k ]_ D )
102	101	rexlimiva 3028	. . . . . . 7 |- ( E. n e. C w e. ( { n } X. [_ n / k ]_ D ) -> ( 2nd ` w ) e. [_ ( 1st ` w ) / k ]_ D )
103	93, 102	syl 17	. . . . . 6 |- ( ( ph /\ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) ) -> ( 2nd ` w ) e. [_ ( 1st ` w ) / k ]_ D )
104		simpl 473	. . . . . . . . . 10 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> n e. C )
105	99, 104	eqeltrd 2701	. . . . . . . . 9 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> ( 1st ` w ) e. C )
106	105	rexlimiva 3028	. . . . . . . 8 |- ( E. n e. C w e. ( { n } X. [_ n / k ]_ D ) -> ( 1st ` w ) e. C )
107	93, 106	syl 17	. . . . . . 7 |- ( ( ph /\ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) ) -> ( 1st ` w ) e. C )
108		simpl 473	. . . . . . . . . 10 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> ph )
109	25	nfcri 2758	. . . . . . . . . . . 12 |- F/ j n e. [_ m / j ]_ B
110	72	equcomd 1946	. . . . . . . . . . . . . . . . 17 |- ( m e. { j } -> j = m )
111	110, 28	syl 17	. . . . . . . . . . . . . . . 16 |- ( m e. { j } -> B = [_ m / j ]_ B )
112	111	eleq2d 2687	. . . . . . . . . . . . . . 15 |- ( m e. { j } -> ( n e. B <-> n e. [_ m / j ]_ B ) )
113	112	biimpa 501	. . . . . . . . . . . . . 14 |- ( ( m e. { j } /\ n e. B ) -> n e. [_ m / j ]_ B )
114	69, 113	sylbi 207	. . . . . . . . . . . . 13 |- ( <. m , n >. e. ( { j } X. B ) -> n e. [_ m / j ]_ B )
115	114	a1i 11	. . . . . . . . . . . 12 |- ( j e. A -> ( <. m , n >. e. ( { j } X. B ) -> n e. [_ m / j ]_ B ) )
116	109, 115	rexlimi 3024	. . . . . . . . . . 11 |- ( E. j e. A <. m , n >. e. ( { j } X. B ) -> n e. [_ m / j ]_ B )
117	67, 116	syl 17	. . . . . . . . . 10 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> n e. [_ m / j ]_ B )
118		fsumcom2.5	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> E e. CC )
119	118	ralrimivva 2971	. . . . . . . . . . . . 13 |- ( ph -> A. j e. A A. k e. B E e. CC )
120		nfcsb1v 3549	. . . . . . . . . . . . . . . 16 |- F/_ j [_ m / j ]_ E
121	120	nfel1 2779	. . . . . . . . . . . . . . 15 |- F/ j [_ m / j ]_ E e. CC
122	25, 121	nfral 2945	. . . . . . . . . . . . . 14 |- F/ j A. k e. [_ m / j ]_ B [_ m / j ]_ E e. CC
123		csbeq1a 3542	. . . . . . . . . . . . . . . 16 |- ( j = m -> E = [_ m / j ]_ E )
124	123	eleq1d 2686	. . . . . . . . . . . . . . 15 |- ( j = m -> ( E e. CC <-> [_ m / j ]_ E e. CC ) )
125	28, 124	raleqbidv 3152	. . . . . . . . . . . . . 14 |- ( j = m -> ( A. k e. B E e. CC <-> A. k e. [_ m / j ]_ B [_ m / j ]_ E e. CC ) )
126	122, 125	rspc 3303	. . . . . . . . . . . . 13 |- ( m e. A -> ( A. j e. A A. k e. B E e. CC -> A. k e. [_ m / j ]_ B [_ m / j ]_ E e. CC ) )
127	119, 126	mpan9 486	. . . . . . . . . . . 12 |- ( ( ph /\ m e. A ) -> A. k e. [_ m / j ]_ B [_ m / j ]_ E e. CC )
128		nfcsb1v 3549	. . . . . . . . . . . . . 14 |- F/_ k [_ n / k ]_ [_ m / j ]_ E
129	128	nfel1 2779	. . . . . . . . . . . . 13 |- F/ k [_ n / k ]_ [_ m / j ]_ E e. CC
130		csbeq1a 3542	. . . . . . . . . . . . . 14 |- ( k = n -> [_ m / j ]_ E = [_ n / k ]_ [_ m / j ]_ E )
131	130	eleq1d 2686	. . . . . . . . . . . . 13 |- ( k = n -> ( [_ m / j ]_ E e. CC <-> [_ n / k ]_ [_ m / j ]_ E e. CC ) )
132	129, 131	rspc 3303	. . . . . . . . . . . 12 |- ( n e. [_ m / j ]_ B -> ( A. k e. [_ m / j ]_ B [_ m / j ]_ E e. CC -> [_ n / k ]_ [_ m / j ]_ E e. CC ) )
133	127, 132	syl5com 31	. . . . . . . . . . 11 |- ( ( ph /\ m e. A ) -> ( n e. [_ m / j ]_ B -> [_ n / k ]_ [_ m / j ]_ E e. CC ) )
134	133	impr 649	. . . . . . . . . 10 |- ( ( ph /\ ( m e. A /\ n e. [_ m / j ]_ B ) ) -> [_ n / k ]_ [_ m / j ]_ E e. CC )
135	108, 77, 117, 134	syl12anc 1324	. . . . . . . . 9 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> [_ n / k ]_ [_ m / j ]_ E e. CC )
136	135	ralrimivva 2971	. . . . . . . 8 |- ( ph -> A. n e. C A. m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E e. CC )
137	136	adantr 481	. . . . . . 7 |- ( ( ph /\ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) ) -> A. n e. C A. m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E e. CC )
138		csbeq1 3536	. . . . . . . . 9 |- ( n = ( 1st ` w ) -> [_ n / k ]_ D = [_ ( 1st ` w ) / k ]_ D )
139		csbeq1 3536	. . . . . . . . . 10 |- ( n = ( 1st ` w ) -> [_ n / k ]_ [_ m / j ]_ E = [_ ( 1st ` w ) / k ]_ [_ m / j ]_ E )
140	139	eleq1d 2686	. . . . . . . . 9 |- ( n = ( 1st ` w ) -> ( [_ n / k ]_ [_ m / j ]_ E e. CC <-> [_ ( 1st ` w ) / k ]_ [_ m / j ]_ E e. CC ) )
141	138, 140	raleqbidv 3152	. . . . . . . 8 |- ( n = ( 1st ` w ) -> ( A. m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E e. CC <-> A. m e. [_ ( 1st ` w ) / k ]_ D [_ ( 1st ` w ) / k ]_ [_ m / j ]_ E e. CC ) )
142	141	rspcv 3305	. . . . . . 7 |- ( ( 1st ` w ) e. C -> ( A. n e. C A. m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E e. CC -> A. m e. [_ ( 1st ` w ) / k ]_ D [_ ( 1st ` w ) / k ]_ [_ m / j ]_ E e. CC ) )
143	107, 137, 142	sylc 65	. . . . . 6 |- ( ( ph /\ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) ) -> A. m e. [_ ( 1st ` w ) / k ]_ D [_ ( 1st ` w ) / k ]_ [_ m / j ]_ E e. CC )
144		csbeq1 3536	. . . . . . . . 9 |- ( m = ( 2nd ` w ) -> [_ m / j ]_ E = [_ ( 2nd ` w ) / j ]_ E )
145	144	csbeq2dv 3992	. . . . . . . 8 |- ( m = ( 2nd ` w ) -> [_ ( 1st ` w ) / k ]_ [_ m / j ]_ E = [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E )
146	145	eleq1d 2686	. . . . . . 7 |- ( m = ( 2nd ` w ) -> ( [_ ( 1st ` w ) / k ]_ [_ m / j ]_ E e. CC <-> [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E e. CC ) )
147	146	rspcv 3305	. . . . . 6 |- ( ( 2nd ` w ) e. [_ ( 1st ` w ) / k ]_ D -> ( A. m e. [_ ( 1st ` w ) / k ]_ D [_ ( 1st ` w ) / k ]_ [_ m / j ]_ E e. CC -> [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E e. CC ) )
148	103, 143, 147	sylc 65	. . . . 5 |- ( ( ph /\ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) ) -> [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E e. CC )
149	49, 55, 85, 90, 148	fsumcnv 14504	. . . 4 |- ( ph -> sum_ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E = sum_ z e. `' U_ n e. C ( { n } X. [_ n / k ]_ D ) [_ ( 2nd ` z ) / k ]_ [_ ( 1st ` z ) / j ]_ E )
150	41, 149	eqtr4d 2659	. . 3 |- ( ph -> sum_ z e. U_ m e. A ( { m } X. [_ m / j ]_ B ) [_ ( 2nd ` z ) / k ]_ [_ ( 1st ` z ) / j ]_ E = sum_ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E )
151		fsumcom2.3	. . . . . 6 |- ( ( ph /\ j e. A ) -> B e. Fin )
152	151	ralrimiva 2966	. . . . 5 |- ( ph -> A. j e. A B e. Fin )
153	25	nfel1 2779	. . . . . 6 |- F/ j [_ m / j ]_ B e. Fin
154	28	eleq1d 2686	. . . . . 6 |- ( j = m -> ( B e. Fin <-> [_ m / j ]_ B e. Fin ) )
155	153, 154	rspc 3303	. . . . 5 |- ( m e. A -> ( A. j e. A B e. Fin -> [_ m / j ]_ B e. Fin ) )
156	152, 155	mpan9 486	. . . 4 |- ( ( ph /\ m e. A ) -> [_ m / j ]_ B e. Fin )
157	55, 58, 156, 134	fsum2d 14502	. . 3 |- ( ph -> sum_ m e. A sum_ n e. [_ m / j ]_ B [_ n / k ]_ [_ m / j ]_ E = sum_ z e. U_ m e. A ( { m } X. [_ m / j ]_ B ) [_ ( 2nd ` z ) / k ]_ [_ ( 1st ` z ) / j ]_ E )
158	49, 56, 80, 135	fsum2d 14502	. . 3 |- ( ph -> sum_ n e. C sum_ m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E = sum_ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E )
159	150, 157, 158	3eqtr4d 2666	. 2 |- ( ph -> sum_ m e. A sum_ n e. [_ m / j ]_ B [_ n / k ]_ [_ m / j ]_ E = sum_ n e. C sum_ m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E )
160		nfcv 2764	. . 3 |- F/_ m sum_ k e. B E
161		nfcv 2764	. . . . 5 |- F/_ j n
162	161, 120	nfcsb 3551	. . . 4 |- F/_ j [_ n / k ]_ [_ m / j ]_ E
163	25, 162	nfsum 14421	. . 3 |- F/_ j sum_ n e. [_ m / j ]_ B [_ n / k ]_ [_ m / j ]_ E
164		nfcv 2764	. . . . 5 |- F/_ n E
165		nfcsb1v 3549	. . . . 5 |- F/_ k [_ n / k ]_ E
166		csbeq1a 3542	. . . . 5 |- ( k = n -> E = [_ n / k ]_ E )
167	164, 165, 166	cbvsumi 14427	. . . 4 |- sum_ k e. B E = sum_ n e. B [_ n / k ]_ E
168	123	csbeq2dv 3992	. . . . . 6 |- ( j = m -> [_ n / k ]_ E = [_ n / k ]_ [_ m / j ]_ E )
169	168	adantr 481	. . . . 5 |- ( ( j = m /\ n e. B ) -> [_ n / k ]_ E = [_ n / k ]_ [_ m / j ]_ E )
170	28, 169	sumeq12dv 14437	. . . 4 |- ( j = m -> sum_ n e. B [_ n / k ]_ E = sum_ n e. [_ m / j ]_ B [_ n / k ]_ [_ m / j ]_ E )
171	167, 170	syl5eq 2668	. . 3 |- ( j = m -> sum_ k e. B E = sum_ n e. [_ m / j ]_ B [_ n / k ]_ [_ m / j ]_ E )
172	160, 163, 171	cbvsumi 14427	. 2 |- sum_ j e. A sum_ k e. B E = sum_ m e. A sum_ n e. [_ m / j ]_ B [_ n / k ]_ [_ m / j ]_ E
173		nfcv 2764	. . 3 |- F/_ n sum_ j e. D E
174	33, 128	nfsum 14421	. . 3 |- F/_ k sum_ m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E
175		nfcv 2764	. . . . 5 |- F/_ m E
176	175, 120, 123	cbvsumi 14427	. . . 4 |- sum_ j e. D E = sum_ m e. D [_ m / j ]_ E
177	130	adantr 481	. . . . 5 |- ( ( k = n /\ m e. D ) -> [_ m / j ]_ E = [_ n / k ]_ [_ m / j ]_ E )
178	36, 177	sumeq12dv 14437	. . . 4 |- ( k = n -> sum_ m e. D [_ m / j ]_ E = sum_ m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E )
179	176, 178	syl5eq 2668	. . 3 |- ( k = n -> sum_ j e. D E = sum_ m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E )
180	173, 174, 179	cbvsumi 14427	. 2 |- sum_ k e. C sum_ j e. D E = sum_ n e. C sum_ m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E
181	159, 172, 180	3eqtr4g 2681	1 |- ( ph -> sum_ j e. A sum_ k e. B E = sum_ k e. C sum_ j e. D E )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   [_csb 3533   {csn 4177   <.cop 4183   U_ciun 4520   X. cxp 5112   `'ccnv 5113   Rel wrel 5119   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   Fincfn 7955   CCcc 9934   sum_csu 14416

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-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-oadd 7564  df-er 7742  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-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

This theorem is referenced by:  fsumcom  14507  fsum0diag  14509  fsumdvdsdiag  24910  dvdsflsumcom  24914  fsumfldivdiag  24916  logfac2  24942  chpchtsum  24944  logfaclbnd  24947  dchrisum0lem1  25205