Theorem fsumrev 11954

Description: Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)

Hypotheses

Ref	Expression
fsumrev.1	|- ( ph -> K e. ZZ )
fsumrev.2	|- ( ph -> M e. ZZ )
fsumrev.3	|- ( ph -> N e. ZZ )
fsumrev.4	|- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
fsumrev.5	|- ( j = ( K - k ) -> A = B )

Assertion

Ref	Expression
fsumrev	|- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( K - N ) ... ( K - M ) ) B )

Distinct variable groups:   A, k   B, j   j, k, K   j, M, k   j, N, k   ph, j, k

Allowed substitution hints:   A( j)   B( k)

Proof of Theorem fsumrev

Step	Hyp	Ref	Expression
1		fsumrev.5	. 2 |- ( j = ( K - k ) -> A = B )
2		fsumrev.1	. . . 4 |- ( ph -> K e. ZZ )
3		fsumrev.3	. . . 4 |- ( ph -> N e. ZZ )
4	2, 3	zsubcld 9574	. . 3 |- ( ph -> ( K - N ) e. ZZ )
5		fsumrev.2	. . . 4 |- ( ph -> M e. ZZ )
6	2, 5	zsubcld 9574	. . 3 |- ( ph -> ( K - M ) e. ZZ )
7	4, 6	fzfigd 10653	. 2 |- ( ph -> ( ( K - N ) ... ( K - M ) ) e. Fin )
8		eqid 2229	. . 3 |- ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) = ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) )
9	2	adantr 276	. . . 4 |- ( ( ph /\ j e. ( ( K - N ) ... ( K - M ) ) ) -> K e. ZZ )
10		elfzelz 10221	. . . . 5 |- ( j e. ( ( K - N ) ... ( K - M ) ) -> j e. ZZ )
11	10	adantl 277	. . . 4 |- ( ( ph /\ j e. ( ( K - N ) ... ( K - M ) ) ) -> j e. ZZ )
12	9, 11	zsubcld 9574	. . 3 |- ( ( ph /\ j e. ( ( K - N ) ... ( K - M ) ) ) -> ( K - j ) e. ZZ )
13	2	adantr 276	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> K e. ZZ )
14		elfzelz 10221	. . . . 5 |- ( k e. ( M ... N ) -> k e. ZZ )
15	14	adantl 277	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> k e. ZZ )
16	13, 15	zsubcld 9574	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> ( K - k ) e. ZZ )
17		simprr 531	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> k = ( K - j ) )
18		simprl 529	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. ( ( K - N ) ... ( K - M ) ) )
19	5	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> M e. ZZ )
20	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> N e. ZZ )
21	2	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> K e. ZZ )
22	18, 10	syl 14	. . . . . . . 8 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j e. ZZ )
23		fzrev 10280	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ j e. ZZ ) ) -> ( j e. ( ( K - N ) ... ( K - M ) ) <-> ( K - j ) e. ( M ... N ) ) )
24	19, 20, 21, 22, 23	syl22anc 1272	. . . . . . 7 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( j e. ( ( K - N ) ... ( K - M ) ) <-> ( K - j ) e. ( M ... N ) ) )
25	18, 24	mpbid 147	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - j ) e. ( M ... N ) )
26	17, 25	eqeltrd 2306	. . . . 5 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> k e. ( M ... N ) )
27	17	oveq2d 6017	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - k ) = ( K - ( K - j ) ) )
28		zcn 9451	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
29		zcn 9451	. . . . . . . 8 |- ( j e. ZZ -> j e. CC )
30		nncan 8375	. . . . . . . 8 |- ( ( K e. CC /\ j e. CC ) -> ( K - ( K - j ) ) = j )
31	28, 29, 30	syl2an 289	. . . . . . 7 |- ( ( K e. ZZ /\ j e. ZZ ) -> ( K - ( K - j ) ) = j )
32	21, 22, 31	syl2anc 411	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - ( K - j ) ) = j )
33	27, 32	eqtr2d 2263	. . . . 5 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> j = ( K - k ) )
34	26, 33	jca 306	. . . 4 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( k e. ( M ... N ) /\ j = ( K - k ) ) )
35		simprr 531	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> j = ( K - k ) )
36		simprl 529	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. ( M ... N ) )
37	5	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> M e. ZZ )
38	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> N e. ZZ )
39	2	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> K e. ZZ )
40	36, 14	syl 14	. . . . . . . 8 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k e. ZZ )
41		fzrev2 10281	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ k e. ZZ ) ) -> ( k e. ( M ... N ) <-> ( K - k ) e. ( ( K - N ) ... ( K - M ) ) ) )
42	37, 38, 39, 40, 41	syl22anc 1272	. . . . . . 7 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( k e. ( M ... N ) <-> ( K - k ) e. ( ( K - N ) ... ( K - M ) ) ) )
43	36, 42	mpbid 147	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - k ) e. ( ( K - N ) ... ( K - M ) ) )
44	35, 43	eqeltrd 2306	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> j e. ( ( K - N ) ... ( K - M ) ) )
45	35	oveq2d 6017	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - j ) = ( K - ( K - k ) ) )
46		zcn 9451	. . . . . . . 8 |- ( k e. ZZ -> k e. CC )
47		nncan 8375	. . . . . . . 8 |- ( ( K e. CC /\ k e. CC ) -> ( K - ( K - k ) ) = k )
48	28, 46, 47	syl2an 289	. . . . . . 7 |- ( ( K e. ZZ /\ k e. ZZ ) -> ( K - ( K - k ) ) = k )
49	39, 40, 48	syl2anc 411	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - ( K - k ) ) = k )
50	45, 49	eqtr2d 2263	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> k = ( K - j ) )
51	44, 50	jca 306	. . . 4 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) )
52	34, 51	impbida 598	. . 3 |- ( ph -> ( ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) <-> ( k e. ( M ... N ) /\ j = ( K - k ) ) ) )
53	8, 12, 16, 52	f1od 6209	. 2 |- ( ph -> ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) : ( ( K - N ) ... ( K - M ) ) -1-1-onto-> ( M ... N ) )
54		simpr 110	. . 3 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> k e. ( ( K - N ) ... ( K - M ) ) )
55	2	adantr 276	. . . 4 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> K e. ZZ )
56		elfzelz 10221	. . . . 5 |- ( k e. ( ( K - N ) ... ( K - M ) ) -> k e. ZZ )
57	56	adantl 277	. . . 4 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> k e. ZZ )
58	55, 57	zsubcld 9574	. . 3 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> ( K - k ) e. ZZ )
59		oveq2 6009	. . . 4 |- ( j = k -> ( K - j ) = ( K - k ) )
60	59, 8	fvmptg 5710	. . 3 |- ( ( k e. ( ( K - N ) ... ( K - M ) ) /\ ( K - k ) e. ZZ ) -> ( ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) ` k ) = ( K - k ) )
61	54, 58, 60	syl2anc 411	. 2 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> ( ( j e. ( ( K - N ) ... ( K - M ) ) |-> ( K - j ) ) ` k ) = ( K - k ) )
62		fsumrev.4	. 2 |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
63	1, 7, 53, 61, 62	fsumf1o 11901	1 |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( K - N ) ... ( K - M ) ) B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   |-> cmpt 4145   ` cfv 5318  (class class class)co 6001   CCcc 7997   - cmin 8317   ZZcz 9446   ...cfz 10204   sum_csu 11864

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-exp 10761  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-sumdc 11865

This theorem is referenced by:  fisumrev2  11957