Theorem sumrbdc 11153

Description: Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)

Hypotheses

Ref	Expression
isummo.1	|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
isummo.2	|- ( ( ph /\ k e. A ) -> B e. CC )
isumrb.4	|- ( ph -> M e. ZZ )
isumrb.5	|- ( ph -> N e. ZZ )
isumrb.6	|- ( ph -> A C_ ( ZZ>= ` M ) )
isumrb.7	|- ( ph -> A C_ ( ZZ>= ` N ) )
isumrb.mdc	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )
isumrb.ndc	|- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> DECID k e. A )

Assertion

Ref	Expression
sumrbdc	|- ( ph -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )

Distinct variable groups:   A, k   k, N   ph, k   k, M

Allowed substitution hints:   B( k)   C( k)   F( k)

Proof of Theorem sumrbdc

Step	Hyp	Ref	Expression
1		isumrb.5	. . . . 5 |- ( ph -> N e. ZZ )
2	1	adantr 274	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> N e. ZZ )
3		seqex 10225	. . . 4 |- seq M ( + , F ) e. _V
4		climres 11077	. . . 4 |- ( ( N e. ZZ /\ seq M ( + , F ) e. _V ) -> ( ( seq M ( + , F ) |` ( ZZ>= ` N ) ) ~~> C <-> seq M ( + , F ) ~~> C ) )
5	2, 3, 4	sylancl 409	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( ( seq M ( + , F ) |` ( ZZ>= ` N ) ) ~~> C <-> seq M ( + , F ) ~~> C ) )
6		isumrb.7	. . . . 5 |- ( ph -> A C_ ( ZZ>= ` N ) )
7		isummo.1	. . . . . 6 |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
8		isummo.2	. . . . . . 7 |- ( ( ph /\ k e. A ) -> B e. CC )
9	8	adantlr 468	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` M ) ) /\ k e. A ) -> B e. CC )
10		isumrb.mdc	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )
11	10	adantlr 468	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` M ) ) /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )
12		simpr 109	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> N e. ( ZZ>= ` M ) )
13	7, 9, 11, 12	sumrbdclem 11151	. . . . 5 |- ( ( ( ph /\ N e. ( ZZ>= ` M ) ) /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )
14	6, 13	mpidan 419	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )
15	14	breq1d 3939	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( ( seq M ( + , F ) |` ( ZZ>= ` N ) ) ~~> C <-> seq N ( + , F ) ~~> C ) )
16	5, 15	bitr3d 189	. 2 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )
17		isumrb.6	. . . . 5 |- ( ph -> A C_ ( ZZ>= ` M ) )
18	8	adantlr 468	. . . . . 6 |- ( ( ( ph /\ M e. ( ZZ>= ` N ) ) /\ k e. A ) -> B e. CC )
19		isumrb.ndc	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> DECID k e. A )
20	19	adantlr 468	. . . . . 6 |- ( ( ( ph /\ M e. ( ZZ>= ` N ) ) /\ k e. ( ZZ>= ` N ) ) -> DECID k e. A )
21		simpr 109	. . . . . 6 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> M e. ( ZZ>= ` N ) )
22	7, 18, 20, 21	sumrbdclem 11151	. . . . 5 |- ( ( ( ph /\ M e. ( ZZ>= ` N ) ) /\ A C_ ( ZZ>= ` M ) ) -> ( seq N ( + , F ) |` ( ZZ>= ` M ) ) = seq M ( + , F ) )
23	17, 22	mpidan 419	. . . 4 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> ( seq N ( + , F ) |` ( ZZ>= ` M ) ) = seq M ( + , F ) )
24	23	breq1d 3939	. . 3 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> ( ( seq N ( + , F ) |` ( ZZ>= ` M ) ) ~~> C <-> seq M ( + , F ) ~~> C ) )
25		isumrb.4	. . . . 5 |- ( ph -> M e. ZZ )
26	25	adantr 274	. . . 4 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> M e. ZZ )
27		seqex 10225	. . . 4 |- seq N ( + , F ) e. _V
28		climres 11077	. . . 4 |- ( ( M e. ZZ /\ seq N ( + , F ) e. _V ) -> ( ( seq N ( + , F ) |` ( ZZ>= ` M ) ) ~~> C <-> seq N ( + , F ) ~~> C ) )
29	26, 27, 28	sylancl 409	. . 3 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> ( ( seq N ( + , F ) |` ( ZZ>= ` M ) ) ~~> C <-> seq N ( + , F ) ~~> C ) )
30	24, 29	bitr3d 189	. 2 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )
31		uztric 9352	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) )
32	25, 1, 31	syl2anc 408	. 2 |- ( ph -> ( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) )
33	16, 30, 32	mpjaodan 787	1 |- ( ph -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  DECID wdc 819   = wceq 1331   e. wcel 1480   _Vcvv 2686   C_ wss 3071   ifcif 3474   class class class wbr 3929   |-> cmpt 3989   |` cres 4541   ` cfv 5123   CCcc 7623   0cc0 7625   + caddc 7628   ZZcz 9059   ZZ>=cuz 9331   seqcseq 10223   ~~> cli 11052

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 7716  ax-resscn 7717  ax-1cn 7718  ax-1re 7719  ax-icn 7720  ax-addcl 7721  ax-addrcl 7722  ax-mulcl 7723  ax-addcom 7725  ax-addass 7727  ax-distr 7729  ax-i2m1 7730  ax-0lt1 7731  ax-0id 7733  ax-rnegex 7734  ax-cnre 7736  ax-pre-ltirr 7737  ax-pre-ltwlin 7738  ax-pre-lttrn 7739  ax-pre-apti 7740  ax-pre-ltadd 7741

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-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-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-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7807  df-mnf 7808  df-xr 7809  df-ltxr 7810  df-le 7811  df-sub 7940  df-neg 7941  df-inn 8726  df-n0 8983  df-z 9060  df-uz 9332  df-fz 9796  df-fzo 9925  df-seqfrec 10224  df-clim 11053

This theorem is referenced by:  summodc  11157  zsumdc  11158