Theorem fisumcom2 12183

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 )
fisumcom2.fi	|- ( ( ph /\ k e. C ) -> D 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
fisumcom2	|- ( 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 fisumcom2

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

Step	Hyp	Ref	Expression
1		relxp 4879	. . . . . . . . 9 |- Rel ( { j } X. B )
2	1	rgenw 2605	. . . . . . . 8 |- A. j e. A Rel ( { j } X. B )
3		reliun 4893	. . . . . . . 8 |- ( Rel U_ j e. A ( { j } X. B ) <-> A. j e. A Rel ( { j } X. B ) )
4	2, 3	mpbir 146	. . . . . . 7 |- Rel U_ j e. A ( { j } X. B )
5		relcnv 5160	. . . . . . 7 |- Rel `' U_ k e. C ( { k } X. D )
6		ancom 266	. . . . . . . . . . . 12 |- ( ( x = j /\ y = k ) <-> ( y = k /\ x = j ) )
7		vex 2824	. . . . . . . . . . . . 13 |- x e. _V
8		vex 2824	. . . . . . . . . . . . 13 |- y e. _V
9	7, 8	opth 4372	. . . . . . . . . . . 12 |- ( <. x , y >. = <. j , k >. <-> ( x = j /\ y = k ) )
10	8, 7	opth 4372	. . . . . . . . . . . 12 |- ( <. y , x >. = <. k , j >. <-> ( y = k /\ x = j ) )
11	6, 9, 10	3bitr4i 212	. . . . . . . . . . 11 |- ( <. x , y >. = <. j , k >. <-> <. y , x >. = <. k , j >. )
12	11	a1i 9	. . . . . . . . . 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 477	. . . . . . . . 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 1921	. . . . . . . 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 4914	. . . . . . . 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 4957	. . . . . . . . 9 |- ( <. x , y >. e. `' U_ k e. C ( { k } X. D ) <-> <. y , x >. e. U_ k e. C ( { k } X. D ) )
18		eliunxp 4914	. . . . . . . . 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 1716	. . . . . . . . 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 206	. . . . . . . 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 223	. . . . . . 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 4866	. . . . . 6 |- ( ph -> U_ j e. A ( { j } X. B ) = `' U_ k e. C ( { k } X. D ) )
23		nfcv 2392	. . . . . . 7 |- F/_ m ( { j } X. B )
24		nfcv 2392	. . . . . . . 8 |- F/_ j { m }
25		nfcsb1v 3180	. . . . . . . 8 |- F/_ j [_ m / j ]_ B
26	24, 25	nfxp 4796	. . . . . . 7 |- F/_ j ( { m } X. [_ m / j ]_ B )
27		sneq 3716	. . . . . . . 8 |- ( j = m -> { j } = { m } )
28		csbeq1a 3156	. . . . . . . 8 |- ( j = m -> B = [_ m / j ]_ B )
29	27, 28	xpeq12d 4794	. . . . . . 7 |- ( j = m -> ( { j } X. B ) = ( { m } X. [_ m / j ]_ B ) )
30	23, 26, 29	cbviun 4044	. . . . . 6 |- U_ j e. A ( { j } X. B ) = U_ m e. A ( { m } X. [_ m / j ]_ B )
31		nfcv 2392	. . . . . . . 8 |- F/_ n ( { k } X. D )
32		nfcv 2392	. . . . . . . . 9 |- F/_ k { n }
33		nfcsb1v 3180	. . . . . . . . 9 |- F/_ k [_ n / k ]_ D
34	32, 33	nfxp 4796	. . . . . . . 8 |- F/_ k ( { n } X. [_ n / k ]_ D )
35		sneq 3716	. . . . . . . . 9 |- ( k = n -> { k } = { n } )
36		csbeq1a 3156	. . . . . . . . 9 |- ( k = n -> D = [_ n / k ]_ D )
37	35, 36	xpeq12d 4794	. . . . . . . 8 |- ( k = n -> ( { k } X. D ) = ( { n } X. [_ n / k ]_ D ) )
38	31, 34, 37	cbviun 4044	. . . . . . 7 |- U_ k e. C ( { k } X. D ) = U_ n e. C ( { n } X. [_ n / k ]_ D )
39	38	cnveqi 4950	. . . . . 6 |- `' U_ k e. C ( { k } X. D ) = `' U_ n e. C ( { n } X. [_ n / k ]_ D )
40	22, 30, 39	3eqtr3g 2294	. . . . 5 |- ( ph -> U_ m e. A ( { m } X. [_ m / j ]_ B ) = `' U_ n e. C ( { n } X. [_ n / k ]_ D ) )
41	40	sumeq1d 12110	. . . 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 2824	. . . . . . . 8 |- n e. _V
43		vex 2824	. . . . . . . 8 |- m e. _V
44	42, 43	op1std 6372	. . . . . . 7 |- ( w = <. n , m >. -> ( 1st ` w ) = n )
45	44	csbeq1d 3154	. . . . . 6 |- ( w = <. n , m >. -> [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E = [_ n / k ]_ [_ ( 2nd ` w ) / j ]_ E )
46	42, 43	op2ndd 6373	. . . . . . . 8 |- ( w = <. n , m >. -> ( 2nd ` w ) = m )
47	46	csbeq1d 3154	. . . . . . 7 |- ( w = <. n , m >. -> [_ ( 2nd ` w ) / j ]_ E = [_ m / j ]_ E )
48	47	csbeq2dv 3173	. . . . . 6 |- ( w = <. n , m >. -> [_ n / k ]_ [_ ( 2nd ` w ) / j ]_ E = [_ n / k ]_ [_ m / j ]_ E )
49	45, 48	eqtrd 2271	. . . . 5 |- ( w = <. n , m >. -> [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E = [_ n / k ]_ [_ m / j ]_ E )
50	43, 42	op2ndd 6373	. . . . . . 7 |- ( z = <. m , n >. -> ( 2nd ` z ) = n )
51	50	csbeq1d 3154	. . . . . 6 |- ( z = <. m , n >. -> [_ ( 2nd ` z ) / k ]_ [_ ( 1st ` z ) / j ]_ E = [_ n / k ]_ [_ ( 1st ` z ) / j ]_ E )
52	43, 42	op1std 6372	. . . . . . . 8 |- ( z = <. m , n >. -> ( 1st ` z ) = m )
53	52	csbeq1d 3154	. . . . . . 7 |- ( z = <. m , n >. -> [_ ( 1st ` z ) / j ]_ E = [_ m / j ]_ E )
54	53	csbeq2dv 3173	. . . . . 6 |- ( z = <. m , n >. -> [_ n / k ]_ [_ ( 1st ` z ) / j ]_ E = [_ n / k ]_ [_ m / j ]_ E )
55	51, 54	eqtrd 2271	. . . . 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		snfig 7093	. . . . . . . . 9 |- ( n e. _V -> { n } e. Fin )
58	57	elv 2825	. . . . . . . 8 |- { n } e. Fin
59		fisumcom2.fi	. . . . . . . . . 10 |- ( ( ph /\ k e. C ) -> D e. Fin )
60	59	ralrimiva 2623	. . . . . . . . 9 |- ( ph -> A. k e. C D e. Fin )
61	33	nfel1 2403	. . . . . . . . . 10 |- F/ k [_ n / k ]_ D e. Fin
62	36	eleq1d 2307	. . . . . . . . . 10 |- ( k = n -> ( D e. Fin <-> [_ n / k ]_ D e. Fin ) )
63	61, 62	rspc 2923	. . . . . . . . 9 |- ( n e. C -> ( A. k e. C D e. Fin -> [_ n / k ]_ D e. Fin ) )
64	60, 63	mpan9 281	. . . . . . . 8 |- ( ( ph /\ n e. C ) -> [_ n / k ]_ D e. Fin )
65		xpfi 7229	. . . . . . . 8 |- ( ( { n } e. Fin /\ [_ n / k ]_ D e. Fin ) -> ( { n } X. [_ n / k ]_ D ) e. Fin )
66	58, 64, 65	sylancr 418	. . . . . . 7 |- ( ( ph /\ n e. C ) -> ( { n } X. [_ n / k ]_ D ) e. Fin )
67	66	ralrimiva 2623	. . . . . 6 |- ( ph -> A. n e. C ( { n } X. [_ n / k ]_ D ) e. Fin )
68		disjsnxp 6463	. . . . . . 7 |- Disj n e. C ( { n } X. [_ n / k ]_ D )
69	68	a1i 9	. . . . . 6 |- ( ph -> Disj n e. C ( { n } X. [_ n / k ]_ D ) )
70		iunfidisj 7250	. . . . . 6 |- ( ( C e. Fin /\ A. n e. C ( { n } X. [_ n / k ]_ D ) e. Fin /\ Disj n e. C ( { n } X. [_ n / k ]_ D ) ) -> U_ n e. C ( { n } X. [_ n / k ]_ D ) e. Fin )
71	56, 67, 69, 70	syl3anc 1278	. . . . 5 |- ( ph -> U_ n e. C ( { n } X. [_ n / k ]_ D ) e. Fin )
72		reliun 4893	. . . . . . 7 |- ( Rel U_ n e. C ( { n } X. [_ n / k ]_ D ) <-> A. n e. C Rel ( { n } X. [_ n / k ]_ D ) )
73		relxp 4879	. . . . . . . 8 |- Rel ( { n } X. [_ n / k ]_ D )
74	73	a1i 9	. . . . . . 7 |- ( n e. C -> Rel ( { n } X. [_ n / k ]_ D ) )
75	72, 74	mprgbir 2608	. . . . . 6 |- Rel U_ n e. C ( { n } X. [_ n / k ]_ D )
76	75	a1i 9	. . . . 5 |- ( ph -> Rel U_ n e. C ( { n } X. [_ n / k ]_ D ) )
77		csbeq1 3150	. . . . . . . 8 |- ( m = ( 2nd ` w ) -> [_ m / j ]_ E = [_ ( 2nd ` w ) / j ]_ E )
78	77	csbeq2dv 3173	. . . . . . 7 |- ( m = ( 2nd ` w ) -> [_ ( 1st ` w ) / k ]_ [_ m / j ]_ E = [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E )
79	78	eleq1d 2307	. . . . . 6 |- ( m = ( 2nd ` w ) -> ( [_ ( 1st ` w ) / k ]_ [_ m / j ]_ E e. CC <-> [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E e. CC ) )
80		csbeq1 3150	. . . . . . . 8 |- ( n = ( 1st ` w ) -> [_ n / k ]_ D = [_ ( 1st ` w ) / k ]_ D )
81		csbeq1 3150	. . . . . . . . 9 |- ( n = ( 1st ` w ) -> [_ n / k ]_ [_ m / j ]_ E = [_ ( 1st ` w ) / k ]_ [_ m / j ]_ E )
82	81	eleq1d 2307	. . . . . . . 8 |- ( n = ( 1st ` w ) -> ( [_ n / k ]_ [_ m / j ]_ E e. CC <-> [_ ( 1st ` w ) / k ]_ [_ m / j ]_ E e. CC ) )
83	80, 82	raleqbidv 2765	. . . . . . 7 |- ( 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 ) )
84		simpl 109	. . . . . . . . . 10 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> ph )
85	43, 42	opelcnv 4957	. . . . . . . . . . . . . . 15 |- ( <. m , n >. e. `' U_ k e. C ( { k } X. D ) <-> <. n , m >. e. U_ k e. C ( { k } X. D ) )
86	33, 36	opeliunxp2f 6499	. . . . . . . . . . . . . . 15 |- ( <. n , m >. e. U_ k e. C ( { k } X. D ) <-> ( n e. C /\ m e. [_ n / k ]_ D ) )
87	85, 86	sylbbr 136	. . . . . . . . . . . . . 14 |- ( ( n e. C /\ m e. [_ n / k ]_ D ) -> <. m , n >. e. `' U_ k e. C ( { k } X. D ) )
88	87	adantl 277	. . . . . . . . . . . . 13 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> <. m , n >. e. `' U_ k e. C ( { k } X. D ) )
89	22	adantr 276	. . . . . . . . . . . . 13 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> U_ j e. A ( { j } X. B ) = `' U_ k e. C ( { k } X. D ) )
90	88, 89	eleqtrrd 2318	. . . . . . . . . . . 12 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> <. m , n >. e. U_ j e. A ( { j } X. B ) )
91		eliun 4011	. . . . . . . . . . . 12 |- ( <. m , n >. e. U_ j e. A ( { j } X. B ) <-> E. j e. A <. m , n >. e. ( { j } X. B ) )
92	90, 91	sylib 122	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> E. j e. A <. m , n >. e. ( { j } X. B ) )
93		simpr 110	. . . . . . . . . . . . . . . 16 |- ( ( j e. A /\ <. m , n >. e. ( { j } X. B ) ) -> <. m , n >. e. ( { j } X. B ) )
94		opelxp 4799	. . . . . . . . . . . . . . . 16 |- ( <. m , n >. e. ( { j } X. B ) <-> ( m e. { j } /\ n e. B ) )
95	93, 94	sylib 122	. . . . . . . . . . . . . . 15 |- ( ( j e. A /\ <. m , n >. e. ( { j } X. B ) ) -> ( m e. { j } /\ n e. B ) )
96	95	simpld 112	. . . . . . . . . . . . . 14 |- ( ( j e. A /\ <. m , n >. e. ( { j } X. B ) ) -> m e. { j } )
97		elsni 3723	. . . . . . . . . . . . . 14 |- ( m e. { j } -> m = j )
98	96, 97	syl 14	. . . . . . . . . . . . 13 |- ( ( j e. A /\ <. m , n >. e. ( { j } X. B ) ) -> m = j )
99		simpl 109	. . . . . . . . . . . . 13 |- ( ( j e. A /\ <. m , n >. e. ( { j } X. B ) ) -> j e. A )
100	98, 99	eqeltrd 2315	. . . . . . . . . . . 12 |- ( ( j e. A /\ <. m , n >. e. ( { j } X. B ) ) -> m e. A )
101	100	rexlimiva 2663	. . . . . . . . . . 11 |- ( E. j e. A <. m , n >. e. ( { j } X. B ) -> m e. A )
102	92, 101	syl 14	. . . . . . . . . 10 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> m e. A )
103	25	nfcri 2386	. . . . . . . . . . . 12 |- F/ j n e. [_ m / j ]_ B
104	97	equcomd 1759	. . . . . . . . . . . . . . . . 17 |- ( m e. { j } -> j = m )
105	104, 28	syl 14	. . . . . . . . . . . . . . . 16 |- ( m e. { j } -> B = [_ m / j ]_ B )
106	105	eleq2d 2308	. . . . . . . . . . . . . . 15 |- ( m e. { j } -> ( n e. B <-> n e. [_ m / j ]_ B ) )
107	106	biimpa 296	. . . . . . . . . . . . . 14 |- ( ( m e. { j } /\ n e. B ) -> n e. [_ m / j ]_ B )
108	94, 107	sylbi 121	. . . . . . . . . . . . 13 |- ( <. m , n >. e. ( { j } X. B ) -> n e. [_ m / j ]_ B )
109	108	a1i 9	. . . . . . . . . . . 12 |- ( j e. A -> ( <. m , n >. e. ( { j } X. B ) -> n e. [_ m / j ]_ B ) )
110	103, 109	rexlimi 2661	. . . . . . . . . . 11 |- ( E. j e. A <. m , n >. e. ( { j } X. B ) -> n e. [_ m / j ]_ B )
111	92, 110	syl 14	. . . . . . . . . 10 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> n e. [_ m / j ]_ B )
112		fsumcom2.5	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> E e. CC )
113	112	ralrimivva 2632	. . . . . . . . . . . . 13 |- ( ph -> A. j e. A A. k e. B E e. CC )
114		nfcsb1v 3180	. . . . . . . . . . . . . . . 16 |- F/_ j [_ m / j ]_ E
115	114	nfel1 2403	. . . . . . . . . . . . . . 15 |- F/ j [_ m / j ]_ E e. CC
116	25, 115	nfralxy 2588	. . . . . . . . . . . . . 14 |- F/ j A. k e. [_ m / j ]_ B [_ m / j ]_ E e. CC
117		csbeq1a 3156	. . . . . . . . . . . . . . . 16 |- ( j = m -> E = [_ m / j ]_ E )
118	117	eleq1d 2307	. . . . . . . . . . . . . . 15 |- ( j = m -> ( E e. CC <-> [_ m / j ]_ E e. CC ) )
119	28, 118	raleqbidv 2765	. . . . . . . . . . . . . 14 |- ( j = m -> ( A. k e. B E e. CC <-> A. k e. [_ m / j ]_ B [_ m / j ]_ E e. CC ) )
120	116, 119	rspc 2923	. . . . . . . . . . . . 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 ) )
121	113, 120	mpan9 281	. . . . . . . . . . . 12 |- ( ( ph /\ m e. A ) -> A. k e. [_ m / j ]_ B [_ m / j ]_ E e. CC )
122		nfcsb1v 3180	. . . . . . . . . . . . . 14 |- F/_ k [_ n / k ]_ [_ m / j ]_ E
123	122	nfel1 2403	. . . . . . . . . . . . 13 |- F/ k [_ n / k ]_ [_ m / j ]_ E e. CC
124		csbeq1a 3156	. . . . . . . . . . . . . 14 |- ( k = n -> [_ m / j ]_ E = [_ n / k ]_ [_ m / j ]_ E )
125	124	eleq1d 2307	. . . . . . . . . . . . 13 |- ( k = n -> ( [_ m / j ]_ E e. CC <-> [_ n / k ]_ [_ m / j ]_ E e. CC ) )
126	123, 125	rspc 2923	. . . . . . . . . . . 12 |- ( n e. [_ m / j ]_ B -> ( A. k e. [_ m / j ]_ B [_ m / j ]_ E e. CC -> [_ n / k ]_ [_ m / j ]_ E e. CC ) )
127	121, 126	syl5com 29	. . . . . . . . . . 11 |- ( ( ph /\ m e. A ) -> ( n e. [_ m / j ]_ B -> [_ n / k ]_ [_ m / j ]_ E e. CC ) )
128	127	impr 379	. . . . . . . . . 10 |- ( ( ph /\ ( m e. A /\ n e. [_ m / j ]_ B ) ) -> [_ n / k ]_ [_ m / j ]_ E e. CC )
129	84, 102, 111, 128	syl12anc 1276	. . . . . . . . 9 |- ( ( ph /\ ( n e. C /\ m e. [_ n / k ]_ D ) ) -> [_ n / k ]_ [_ m / j ]_ E e. CC )
130	129	ralrimivva 2632	. . . . . . . 8 |- ( ph -> A. n e. C A. m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E e. CC )
131	130	adantr 276	. . . . . . 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 )
132		simpr 110	. . . . . . . . 9 |- ( ( ph /\ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) ) -> w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) )
133		eliun 4011	. . . . . . . . 9 |- ( w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) <-> E. n e. C w e. ( { n } X. [_ n / k ]_ D ) )
134	132, 133	sylib 122	. . . . . . . 8 |- ( ( ph /\ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) ) -> E. n e. C w e. ( { n } X. [_ n / k ]_ D ) )
135		xp1st 6389	. . . . . . . . . . . 12 |- ( w e. ( { n } X. [_ n / k ]_ D ) -> ( 1st ` w ) e. { n } )
136	135	adantl 277	. . . . . . . . . . 11 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> ( 1st ` w ) e. { n } )
137		elsni 3723	. . . . . . . . . . 11 |- ( ( 1st ` w ) e. { n } -> ( 1st ` w ) = n )
138	136, 137	syl 14	. . . . . . . . . 10 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> ( 1st ` w ) = n )
139		simpl 109	. . . . . . . . . 10 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> n e. C )
140	138, 139	eqeltrd 2315	. . . . . . . . 9 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> ( 1st ` w ) e. C )
141	140	rexlimiva 2663	. . . . . . . 8 |- ( E. n e. C w e. ( { n } X. [_ n / k ]_ D ) -> ( 1st ` w ) e. C )
142	134, 141	syl 14	. . . . . . 7 |- ( ( ph /\ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) ) -> ( 1st ` w ) e. C )
143	83, 131, 142	rspcdva 2934	. . . . . 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		xp2nd 6390	. . . . . . . . . 10 |- ( w e. ( { n } X. [_ n / k ]_ D ) -> ( 2nd ` w ) e. [_ n / k ]_ D )
145	144	adantl 277	. . . . . . . . 9 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> ( 2nd ` w ) e. [_ n / k ]_ D )
146	138	csbeq1d 3154	. . . . . . . . 9 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> [_ ( 1st ` w ) / k ]_ D = [_ n / k ]_ D )
147	145, 146	eleqtrrd 2318	. . . . . . . 8 |- ( ( n e. C /\ w e. ( { n } X. [_ n / k ]_ D ) ) -> ( 2nd ` w ) e. [_ ( 1st ` w ) / k ]_ D )
148	147	rexlimiva 2663	. . . . . . 7 |- ( E. n e. C w e. ( { n } X. [_ n / k ]_ D ) -> ( 2nd ` w ) e. [_ ( 1st ` w ) / k ]_ D )
149	134, 148	syl 14	. . . . . 6 |- ( ( ph /\ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) ) -> ( 2nd ` w ) e. [_ ( 1st ` w ) / k ]_ D )
150	79, 143, 149	rspcdva 2934	. . . . 5 |- ( ( ph /\ w e. U_ n e. C ( { n } X. [_ n / k ]_ D ) ) -> [_ ( 1st ` w ) / k ]_ [_ ( 2nd ` w ) / j ]_ E e. CC )
151	49, 55, 71, 76, 150	fsumcnv 12182	. . . 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 )
152	41, 151	eqtr4d 2274	. . 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 )
153		fsumcom2.1	. . . 4 |- ( ph -> A e. Fin )
154		fsumcom2.3	. . . . . 6 |- ( ( ph /\ j e. A ) -> B e. Fin )
155	154	ralrimiva 2623	. . . . 5 |- ( ph -> A. j e. A B e. Fin )
156	25	nfel1 2403	. . . . . 6 |- F/ j [_ m / j ]_ B e. Fin
157	28	eleq1d 2307	. . . . . 6 |- ( j = m -> ( B e. Fin <-> [_ m / j ]_ B e. Fin ) )
158	156, 157	rspc 2923	. . . . 5 |- ( m e. A -> ( A. j e. A B e. Fin -> [_ m / j ]_ B e. Fin ) )
159	155, 158	mpan9 281	. . . 4 |- ( ( ph /\ m e. A ) -> [_ m / j ]_ B e. Fin )
160	55, 153, 159, 128	fsum2d 12180	. . 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 )
161	49, 56, 64, 129	fsum2d 12180	. . 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 )
162	152, 160, 161	3eqtr4d 2281	. 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 )
163		nfcv 2392	. . 3 |- F/_ m sum_ k e. B E
164		nfcv 2392	. . . . 5 |- F/_ j n
165	164, 114	nfcsb 3185	. . . 4 |- F/_ j [_ n / k ]_ [_ m / j ]_ E
166	25, 165	nfsum 12101	. . 3 |- F/_ j sum_ n e. [_ m / j ]_ B [_ n / k ]_ [_ m / j ]_ E
167		nfcv 2392	. . . . 5 |- F/_ n E
168		nfcsb1v 3180	. . . . 5 |- F/_ k [_ n / k ]_ E
169		csbeq1a 3156	. . . . 5 |- ( k = n -> E = [_ n / k ]_ E )
170	167, 168, 169	cbvsumi 12106	. . . 4 |- sum_ k e. B E = sum_ n e. B [_ n / k ]_ E
171	117	csbeq2dv 3173	. . . . . 6 |- ( j = m -> [_ n / k ]_ E = [_ n / k ]_ [_ m / j ]_ E )
172	171	adantr 276	. . . . 5 |- ( ( j = m /\ n e. B ) -> [_ n / k ]_ E = [_ n / k ]_ [_ m / j ]_ E )
173	28, 172	sumeq12dv 12116	. . . 4 |- ( j = m -> sum_ n e. B [_ n / k ]_ E = sum_ n e. [_ m / j ]_ B [_ n / k ]_ [_ m / j ]_ E )
174	170, 173	eqtrid 2283	. . 3 |- ( j = m -> sum_ k e. B E = sum_ n e. [_ m / j ]_ B [_ n / k ]_ [_ m / j ]_ E )
175	163, 166, 174	cbvsumi 12106	. 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
176		nfcv 2392	. . 3 |- F/_ n sum_ j e. D E
177	33, 122	nfsum 12101	. . 3 |- F/_ k sum_ m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E
178		nfcv 2392	. . . . 5 |- F/_ m E
179	178, 114, 117	cbvsumi 12106	. . . 4 |- sum_ j e. D E = sum_ m e. D [_ m / j ]_ E
180	124	adantr 276	. . . . 5 |- ( ( k = n /\ m e. D ) -> [_ m / j ]_ E = [_ n / k ]_ [_ m / j ]_ E )
181	36, 180	sumeq12dv 12116	. . . 4 |- ( k = n -> sum_ m e. D [_ m / j ]_ E = sum_ m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E )
182	179, 181	eqtrid 2283	. . 3 |- ( k = n -> sum_ j e. D E = sum_ m e. [_ n / k ]_ D [_ n / k ]_ [_ m / j ]_ E )
183	176, 177, 182	cbvsumi 12106	. 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
184	162, 175, 183	3eqtr4g 2296	1 |- ( ph -> sum_ j e. A sum_ k e. B E = sum_ k e. C sum_ j e. D E )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   _Vcvv 2821   [_csb 3147   {csn 3705   <.cop 3708   U_ciun 4007  Disj wdisj 4101   X. cxp 4767   `'ccnv 4768   Rel wrel 4774   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   Fincfn 7012   CCcc 8167   sum_csu 12097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-disj 4102  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098

This theorem is referenced by:  fsumcom  12184  fisum0diag  12186