Theorem fsumcnv 11447

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 3068	. . . 4 |- ( x = <. ( 2nd ` y ) , ( 1st ` y ) >. -> B = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
2		2ndexg 6171	. . . . . 6 |- ( y e. _V -> ( 2nd ` y ) e. _V )
3	2	elv 2743	. . . . 5 |- ( 2nd ` y ) e. _V
4		1stexg 6170	. . . . . 6 |- ( y e. _V -> ( 1st ` y ) e. _V )
5	4	elv 2743	. . . . 5 |- ( 1st ` y ) e. _V
6		vex 2742	. . . . . . . 8 |- j e. _V
7		vex 2742	. . . . . . . 8 |- k e. _V
8	6, 7	opex 4231	. . . . . . 7 |- <. j , k >. e. _V
9		fsumcnv.1	. . . . . . 7 |- ( x = <. j , k >. -> B = D )
10	8, 9	csbie 3104	. . . . . 6 |- [_ <. j , k >. / x ]_ B = D
11		opeq12 3782	. . . . . . 7 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> <. j , k >. = <. ( 2nd ` y ) , ( 1st ` y ) >. )
12	11	csbeq1d 3066	. . . . . 6 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> [_ <. j , k >. / x ]_ B = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
13	10, 12	eqtr3id 2224	. . . . 5 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> D = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
14	3, 5, 13	csbie2 3108	. . . 4 |- [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B
15	1, 14	eqtr4di 2228	. . 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 6942	. . . 4 |- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin )
19	16, 17, 18	syl2anc 411	. . 3 |- ( ph -> `' A e. Fin )
20		relcnv 5008	. . . . 5 |- Rel `' A
21		cnvf1o 6228	. . . . 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 5081	. . . . . 6 |- ( Rel A <-> `' `' A = A )
24	16, 23	sylib 122	. . . . 5 |- ( ph -> `' `' A = A )
25		f1oeq3 5453	. . . . 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 6184	. . . . . . 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 5521	. . . . 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 2255	. . . . . 6 |- ( y e. `' A -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. `' A )
33		sneq 3605	. . . . . . . . . 10 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> { z } = { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
34	33	cnveqd 4805	. . . . . . . . 9 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> `' { z } = `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
35	34	unieqd 3822	. . . . . . . 8 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> U. `' { z } = U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
36		opswapg 5117	. . . . . . . . 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 2226	. . . . . . 7 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> U. `' { z } = <. ( 2nd ` y ) , ( 1st ` y ) >. )
39		eqid 2177	. . . . . . 7 |- ( z e. `' A |-> U. `' { z } ) = ( z e. `' A |-> U. `' { z } )
40	3, 5	opex 4231	. . . . . . 7 |- <. ( 2nd ` y ) , ( 1st ` y ) >. e. _V
41	38, 39, 40	fvmpt 5595	. . . . . 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 2210	. . . 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 11400	. 2 |- ( ph -> sum_ x e. A B = sum_ y e. `' A [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D )
47		csbeq1a 3068	. . . . 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 4231	. . . . . . 7 |- <. k , j >. e. _V
50		fsumcnv.2	. . . . . . 7 |- ( y = <. k , j >. -> C = D )
51	49, 50	csbie 3104	. . . . . 6 |- [_ <. k , j >. / y ]_ C = D
52		opeq12 3782	. . . . . . . 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 3066	. . . . . 6 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> [_ <. k , j >. / y ]_ C = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
55	51, 54	eqtr3id 2224	. . . . 5 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> D = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
56	3, 5, 55	csbie2 3108	. . . 4 |- [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C
57	48, 56	eqtr4di 2228	. . 3 |- ( y e. `' A -> C = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D )
58	57	sumeq2i 11374	. 2 |- sum_ y e. `' A C = sum_ y e. `' A [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D
59	46, 58	eqtr4di 2228	1 |- ( ph -> sum_ x e. A B = sum_ y e. `' A C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2739   [_csb 3059   {csn 3594   <.cop 3597   U.cuni 3811   |-> cmpt 4066   `'ccnv 4627   Rel wrel 4633   -1-1-onto->wf1o 5217   ` cfv 5218   1stc1st 6141   2ndc2nd 6142   Fincfn 6742   CCcc 7811   sum_csu 11363

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-oadd 6423  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-seqfrec 10448  df-exp 10522  df-ihash 10758  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289  df-sumdc 11364

This theorem is referenced by:  fisumcom2  11448