Theorem fsumcnv 11206

Description: Transform a region of summation by using the converse operation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Hypotheses

Ref	Expression
fsumcnv.1	|- ( x = <. j , k >. -> B = D )
fsumcnv.2	|- ( y = <. k , j >. -> C = D )
fsumcnv.3	|- ( ph -> A e. Fin )
fsumcnv.4	|- ( ph -> Rel A )
fsumcnv.5	|- ( ( ph /\ x e. A ) -> B e. CC )

Assertion

Ref	Expression
fsumcnv	|- ( ph -> sum_ x e. A B = sum_ y e. `' A C )

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

Allowed substitution hints:   ph( j, k)   A( j, k)   B( x)   C( y)   D( j, k)

Proof of Theorem fsumcnv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1a 3012	. . . 4 |- ( x = <. ( 2nd ` y ) , ( 1st ` y ) >. -> B = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
2		2ndexg 6066	. . . . . 6 |- ( y e. _V -> ( 2nd ` y ) e. _V )
3	2	elv 2690	. . . . 5 |- ( 2nd ` y ) e. _V
4		1stexg 6065	. . . . . 6 |- ( y e. _V -> ( 1st ` y ) e. _V )
5	4	elv 2690	. . . . 5 |- ( 1st ` y ) e. _V
6		vex 2689	. . . . . . . 8 |- j e. _V
7		vex 2689	. . . . . . . 8 |- k e. _V
8	6, 7	opex 4151	. . . . . . 7 |- <. j , k >. e. _V
9		fsumcnv.1	. . . . . . 7 |- ( x = <. j , k >. -> B = D )
10	8, 9	csbie 3045	. . . . . 6 |- [_ <. j , k >. / x ]_ B = D
11		opeq12 3707	. . . . . . 7 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> <. j , k >. = <. ( 2nd ` y ) , ( 1st ` y ) >. )
12	11	csbeq1d 3010	. . . . . 6 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> [_ <. j , k >. / x ]_ B = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
13	10, 12	syl5eqr 2186	. . . . 5 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> D = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
14	3, 5, 13	csbie2 3049	. . . 4 |- [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B
15	1, 14	syl6eqr 2190	. . 3 |- ( x = <. ( 2nd ` y ) , ( 1st ` y ) >. -> B = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D )
16		fsumcnv.4	. . . 4 |- ( ph -> Rel A )
17		fsumcnv.3	. . . 4 |- ( ph -> A e. Fin )
18		relcnvfi 6829	. . . 4 |- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin )
19	16, 17, 18	syl2anc 408	. . 3 |- ( ph -> `' A e. Fin )
20		relcnv 4917	. . . . 5 |- Rel `' A
21		cnvf1o 6122	. . . . 5 |- ( Rel `' A -> ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> `' `' A )
22	20, 21	ax-mp 5	. . . 4 |- ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> `' `' A
23		dfrel2 4989	. . . . . 6 |- ( Rel A <-> `' `' A = A )
24	16, 23	sylib 121	. . . . 5 |- ( ph -> `' `' A = A )
25		f1oeq3 5358	. . . . 5 |- ( `' `' A = A -> ( ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> `' `' A <-> ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> A ) )
26	24, 25	syl 14	. . . 4 |- ( ph -> ( ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> `' `' A <-> ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> A ) )
27	22, 26	mpbii 147	. . 3 |- ( ph -> ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> A )
28		1st2nd 6079	. . . . . . 7 |- ( ( Rel `' A /\ y e. `' A ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
29	20, 28	mpan 420	. . . . . 6 |- ( y e. `' A -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
30	29	fveq2d 5425	. . . . 5 |- ( y e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` y ) = ( ( z e. `' A |-> U. `' { z } ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) )
31		id 19	. . . . . . 7 |- ( y e. `' A -> y e. `' A )
32	29, 31	eqeltrrd 2217	. . . . . 6 |- ( y e. `' A -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. `' A )
33		sneq 3538	. . . . . . . . . 10 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> { z } = { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
34	33	cnveqd 4715	. . . . . . . . 9 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> `' { z } = `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
35	34	unieqd 3747	. . . . . . . 8 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> U. `' { z } = U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
36		opswapg 5025	. . . . . . . . 9 |- ( ( ( 1st ` y ) e. _V /\ ( 2nd ` y ) e. _V ) -> U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } = <. ( 2nd ` y ) , ( 1st ` y ) >. )
37	5, 3, 36	mp2an 422	. . . . . . . 8 |- U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } = <. ( 2nd ` y ) , ( 1st ` y ) >.
38	35, 37	syl6eq 2188	. . . . . . 7 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> U. `' { z } = <. ( 2nd ` y ) , ( 1st ` y ) >. )
39		eqid 2139	. . . . . . 7 |- ( z e. `' A |-> U. `' { z } ) = ( z e. `' A |-> U. `' { z } )
40	3, 5	opex 4151	. . . . . . 7 |- <. ( 2nd ` y ) , ( 1st ` y ) >. e. _V
41	38, 39, 40	fvmpt 5498	. . . . . 6 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) = <. ( 2nd ` y ) , ( 1st ` y ) >. )
42	32, 41	syl 14	. . . . 5 |- ( y e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) = <. ( 2nd ` y ) , ( 1st ` y ) >. )
43	30, 42	eqtrd 2172	. . . 4 |- ( y e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` y ) = <. ( 2nd ` y ) , ( 1st ` y ) >. )
44	43	adantl 275	. . 3 |- ( ( ph /\ y e. `' A ) -> ( ( z e. `' A |-> U. `' { z } ) ` y ) = <. ( 2nd ` y ) , ( 1st ` y ) >. )
45		fsumcnv.5	. . 3 |- ( ( ph /\ x e. A ) -> B e. CC )
46	15, 19, 27, 44, 45	fsumf1o 11159	. 2 |- ( ph -> sum_ x e. A B = sum_ y e. `' A [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D )
47		csbeq1a 3012	. . . . 5 |- ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. -> C = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
48	29, 47	syl 14	. . . 4 |- ( y e. `' A -> C = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
49	7, 6	opex 4151	. . . . . . 7 |- <. k , j >. e. _V
50		fsumcnv.2	. . . . . . 7 |- ( y = <. k , j >. -> C = D )
51	49, 50	csbie 3045	. . . . . 6 |- [_ <. k , j >. / y ]_ C = D
52		opeq12 3707	. . . . . . . 8 |- ( ( k = ( 1st ` y ) /\ j = ( 2nd ` y ) ) -> <. k , j >. = <. ( 1st ` y ) , ( 2nd ` y ) >. )
53	52	ancoms 266	. . . . . . 7 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> <. k , j >. = <. ( 1st ` y ) , ( 2nd ` y ) >. )
54	53	csbeq1d 3010	. . . . . 6 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> [_ <. k , j >. / y ]_ C = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
55	51, 54	syl5eqr 2186	. . . . 5 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> D = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
56	3, 5, 55	csbie2 3049	. . . 4 |- [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C
57	48, 56	syl6eqr 2190	. . 3 |- ( y e. `' A -> C = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D )
58	57	sumeq2i 11133	. 2 |- sum_ y e. `' A C = sum_ y e. `' A [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D
59	46, 58	syl6eqr 2190	1 |- ( ph -> sum_ x e. A B = sum_ y e. `' A C )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   _Vcvv 2686   [_csb 3003   {csn 3527   <.cop 3530   U.cuni 3736   |-> cmpt 3989   `'ccnv 4538   Rel wrel 4544   -1-1-onto->wf1o 5122   ` cfv 5123   1stc1st 6036   2ndc2nd 6037   Fincfn 6634   CCcc 7618   sum_csu 11122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123

This theorem is referenced by:  fisumcom2  11207