Theorem sumrbdc 11140

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 10213	. . . 4 |- seq M ( + , F ) e. _V
4		climres 11065	. . . 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 11138	. . . . 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 3934	. . 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 11138	. . . . 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 3934	. . 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 10213	. . . 4 |- seq N ( + , F ) e. _V
28		climres 11065	. . . 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 9340	. . 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 2681   C_ wss 3066   ifcif 3469   class class class wbr 3924   |-> cmpt 3984   |` cres 4536   ` cfv 5118   CCcc 7611   0cc0 7613   + caddc 7616   ZZcz 9047   ZZ>=cuz 9319   seqcseq 10211   ~~> cli 11040

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 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729

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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-fz 9784  df-fzo 9913  df-seqfrec 10212  df-clim 11041

This theorem is referenced by:  summodc  11145  zsumdc  11146