Theorem sumrbdc 11890

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 276	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> N e. ZZ )
3		seqex 10671	. . . 4 |- seq M ( + , F ) e. _V
4		climres 11814	. . . 4 |- ( ( N e. ZZ /\ seq M ( + , F ) e. _V ) -> ( ( seq M ( + , F ) |` ( ZZ>= ` N ) ) ~~> C <-> seq M ( + , F ) ~~> C ) )
5	2, 3, 4	sylancl 413	. . 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 477	. . . . . 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 477	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` M ) ) /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )
12		simpr 110	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> N e. ( ZZ>= ` M ) )
13	7, 9, 11, 12	sumrbdclem 11888	. . . . 5 |- ( ( ( ph /\ N e. ( ZZ>= ` M ) ) /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )
14	6, 13	mpidan 423	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )
15	14	breq1d 4093	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( ( seq M ( + , F ) |` ( ZZ>= ` N ) ) ~~> C <-> seq N ( + , F ) ~~> C ) )
16	5, 15	bitr3d 190	. 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 477	. . . . . 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 477	. . . . . 6 |- ( ( ( ph /\ M e. ( ZZ>= ` N ) ) /\ k e. ( ZZ>= ` N ) ) -> DECID k e. A )
21		simpr 110	. . . . . 6 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> M e. ( ZZ>= ` N ) )
22	7, 18, 20, 21	sumrbdclem 11888	. . . . 5 |- ( ( ( ph /\ M e. ( ZZ>= ` N ) ) /\ A C_ ( ZZ>= ` M ) ) -> ( seq N ( + , F ) |` ( ZZ>= ` M ) ) = seq M ( + , F ) )
23	17, 22	mpidan 423	. . . 4 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> ( seq N ( + , F ) |` ( ZZ>= ` M ) ) = seq M ( + , F ) )
24	23	breq1d 4093	. . 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 276	. . . 4 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> M e. ZZ )
27		seqex 10671	. . . 4 |- seq N ( + , F ) e. _V
28		climres 11814	. . . 4 |- ( ( M e. ZZ /\ seq N ( + , F ) e. _V ) -> ( ( seq N ( + , F ) |` ( ZZ>= ` M ) ) ~~> C <-> seq N ( + , F ) ~~> C ) )
29	26, 27, 28	sylancl 413	. . 3 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> ( ( seq N ( + , F ) |` ( ZZ>= ` M ) ) ~~> C <-> seq N ( + , F ) ~~> C ) )
30	24, 29	bitr3d 190	. 2 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )
31		uztric 9744	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) )
32	25, 1, 31	syl2anc 411	. 2 |- ( ph -> ( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) )
33	16, 30, 32	mpjaodan 803	1 |- ( ph -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   ifcif 3602   class class class wbr 4083   |-> cmpt 4145   |` cres 4721   ` cfv 5318   CCcc 7997   0cc0 7999   + caddc 8002   ZZcz 9446   ZZ>=cuz 9722   seqcseq 10669   ~~> cli 11789

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-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115

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-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-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-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-clim 11790

This theorem is referenced by:  summodc  11894  zsumdc  11895