Theorem fsumcnv 12127

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 3149	. . . 4 |- ( x = <. ( 2nd ` y ) , ( 1st ` y ) >. -> B = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
2		2ndexg 6364	. . . . . 6 |- ( y e. _V -> ( 2nd ` y ) e. _V )
3	2	elv 2819	. . . . 5 |- ( 2nd ` y ) e. _V
4		1stexg 6363	. . . . . 6 |- ( y e. _V -> ( 1st ` y ) e. _V )
5	4	elv 2819	. . . . 5 |- ( 1st ` y ) e. _V
6		vex 2818	. . . . . . . 8 |- j e. _V
7		vex 2818	. . . . . . . 8 |- k e. _V
8	6, 7	opex 4347	. . . . . . 7 |- <. j , k >. e. _V
9		fsumcnv.1	. . . . . . 7 |- ( x = <. j , k >. -> B = D )
10	8, 9	csbie 3186	. . . . . 6 |- [_ <. j , k >. / x ]_ B = D
11		opeq12 3887	. . . . . . 7 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> <. j , k >. = <. ( 2nd ` y ) , ( 1st ` y ) >. )
12	11	csbeq1d 3147	. . . . . 6 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> [_ <. j , k >. / x ]_ B = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
13	10, 12	eqtr3id 2281	. . . . 5 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> D = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
14	3, 5, 13	csbie2 3190	. . . 4 |- [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B
15	1, 14	eqtr4di 2285	. . 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 7210	. . . 4 |- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin )
19	16, 17, 18	syl2anc 411	. . 3 |- ( ph -> `' A e. Fin )
20		relcnv 5142	. . . . 5 |- Rel `' A
21		cnvf1o 6423	. . . . 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 5215	. . . . . 6 |- ( Rel A <-> `' `' A = A )
24	16, 23	sylib 122	. . . . 5 |- ( ph -> `' `' A = A )
25		f1oeq3 5606	. . . . 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 148	. . 3 |- ( ph -> ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> A )
28		1st2nd 6377	. . . . . . 7 |- ( ( Rel `' A /\ y e. `' A ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
29	20, 28	mpan 424	. . . . . 6 |- ( y e. `' A -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
30	29	fveq2d 5676	. . . . 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 2312	. . . . . 6 |- ( y e. `' A -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. `' A )
33		sneq 3702	. . . . . . . . . 10 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> { z } = { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
34	33	cnveqd 4933	. . . . . . . . 9 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> `' { z } = `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
35	34	unieqd 3927	. . . . . . . 8 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> U. `' { z } = U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
36		opswapg 5251	. . . . . . . . 9 |- ( ( ( 1st ` y ) e. _V /\ ( 2nd ` y ) e. _V ) -> U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } = <. ( 2nd ` y ) , ( 1st ` y ) >. )
37	5, 3, 36	mp2an 426	. . . . . . . 8 |- U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } = <. ( 2nd ` y ) , ( 1st ` y ) >.
38	35, 37	eqtrdi 2283	. . . . . . 7 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> U. `' { z } = <. ( 2nd ` y ) , ( 1st ` y ) >. )
39		eqid 2234	. . . . . . 7 |- ( z e. `' A |-> U. `' { z } ) = ( z e. `' A |-> U. `' { z } )
40	3, 5	opex 4347	. . . . . . 7 |- <. ( 2nd ` y ) , ( 1st ` y ) >. e. _V
41	38, 39, 40	fvmpt 5756	. . . . . 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 2267	. . . 4 |- ( y e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` y ) = <. ( 2nd ` y ) , ( 1st ` y ) >. )
44	43	adantl 277	. . 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 12080	. 2 |- ( ph -> sum_ x e. A B = sum_ y e. `' A [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D )
47		csbeq1a 3149	. . . . 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 4347	. . . . . . 7 |- <. k , j >. e. _V
50		fsumcnv.2	. . . . . . 7 |- ( y = <. k , j >. -> C = D )
51	49, 50	csbie 3186	. . . . . 6 |- [_ <. k , j >. / y ]_ C = D
52		opeq12 3887	. . . . . . . 8 |- ( ( k = ( 1st ` y ) /\ j = ( 2nd ` y ) ) -> <. k , j >. = <. ( 1st ` y ) , ( 2nd ` y ) >. )
53	52	ancoms 268	. . . . . . 7 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> <. k , j >. = <. ( 1st ` y ) , ( 2nd ` y ) >. )
54	53	csbeq1d 3147	. . . . . 6 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> [_ <. k , j >. / y ]_ C = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
55	51, 54	eqtr3id 2281	. . . . 5 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> D = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
56	3, 5, 55	csbie2 3190	. . . 4 |- [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C
57	48, 56	eqtr4di 2285	. . 3 |- ( y e. `' A -> C = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D )
58	57	sumeq2i 12053	. 2 |- sum_ y e. `' A C = sum_ y e. `' A [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D
59	46, 58	eqtr4di 2285	1 |- ( ph -> sum_ x e. A B = sum_ y e. `' A C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   [_csb 3140   {csn 3691   <.cop 3694   U.cuni 3916   |-> cmpt 4173   `'ccnv 4750   Rel wrel 4756   -1-1-onto->wf1o 5353   ` cfv 5354   1stc1st 6334   2ndc2nd 6335   Fincfn 6977   CCcc 8127   sum_csu 12042

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-fz 10346  df-fzo 10481  df-seqfrec 10814  df-exp 10905  df-ihash 11143  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-clim 11968  df-sumdc 12043

This theorem is referenced by:  fisumcom2  12128