Theorem sumrbdc 11353

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 10415	. . . 4 |- seq M ( + , F ) e. _V
4		climres 11277	. . . 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 11351	. . . . 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 4008	. . 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 11351	. . . . 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 4008	. . 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 10415	. . . 4 |- seq N ( + , F ) e. _V
28		climres 11277	. . . 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 9520	. . 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 798	1 |- ( ph -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   = wceq 1353   e. wcel 2146   _Vcvv 2735   C_ wss 3127   ifcif 3532   class class class wbr 3998   |-> cmpt 4059   |` cres 4622   ` cfv 5208   CCcc 7784   0cc0 7786   + caddc 7789   ZZcz 9224   ZZ>=cuz 9499   seqcseq 10413   ~~> cli 11252

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-inn 8891  df-n0 9148  df-z 9225  df-uz 9500  df-fz 9978  df-fzo 10111  df-seqfrec 10414  df-clim 11253

This theorem is referenced by:  summodc  11357  zsumdc  11358