Theorem fsumrev 11586

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 9444	. . 3 |- ( ph -> ( K - N ) e. ZZ )
5		fsumrev.2	. . . 4 |- ( ph -> M e. ZZ )
6	2, 5	zsubcld 9444	. . 3 |- ( ph -> ( K - M ) e. ZZ )
7	4, 6	fzfigd 10502	. 2 |- ( ph -> ( ( K - N ) ... ( K - M ) ) e. Fin )
8		eqid 2193	. . 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 10091	. . . . 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 9444	. . 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 10091	. . . . 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 9444	. . 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 10150	. . . . . . . 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 1250	. . . . . . 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 2270	. . . . 5 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> k e. ( M ... N ) )
27	17	oveq2d 5934	. . . . . 6 |- ( ( ph /\ ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) ) -> ( K - k ) = ( K - ( K - j ) ) )
28		zcn 9322	. . . . . . . 8 |- ( K e. ZZ -> K e. CC )
29		zcn 9322	. . . . . . . 8 |- ( j e. ZZ -> j e. CC )
30		nncan 8248	. . . . . . . 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 2227	. . . . 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 10151	. . . . . . . 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 1250	. . . . . . 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 2270	. . . . 5 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> j e. ( ( K - N ) ... ( K - M ) ) )
45	35	oveq2d 5934	. . . . . 6 |- ( ( ph /\ ( k e. ( M ... N ) /\ j = ( K - k ) ) ) -> ( K - j ) = ( K - ( K - k ) ) )
46		zcn 9322	. . . . . . . 8 |- ( k e. ZZ -> k e. CC )
47		nncan 8248	. . . . . . . 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 2227	. . . . 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 596	. . 3 |- ( ph -> ( ( j e. ( ( K - N ) ... ( K - M ) ) /\ k = ( K - j ) ) <-> ( k e. ( M ... N ) /\ j = ( K - k ) ) ) )
53	8, 12, 16, 52	f1od 6121	. 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 10091	. . . . 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 9444	. . 3 |- ( ( ph /\ k e. ( ( K - N ) ... ( K - M ) ) ) -> ( K - k ) e. ZZ )
59		oveq2 5926	. . . 4 |- ( j = k -> ( K - j ) = ( K - k ) )
60	59, 8	fvmptg 5633	. . 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 11533	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 1364   e. wcel 2164   |-> cmpt 4090   ` cfv 5254  (class class class)co 5918   CCcc 7870   - cmin 8190   ZZcz 9317   ...cfz 10074   sum_csu 11496

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-exp 10610  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-sumdc 11497

This theorem is referenced by:  fisumrev2  11589