Theorem sumrbdc 11428

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 10486	. . . 4 |- seq M ( + , F ) e. _V
4		climres 11352	. . . 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 11426	. . . . 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 4031	. . 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 11426	. . . . 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 4031	. . 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 10486	. . . 4 |- seq N ( + , F ) e. _V
28		climres 11352	. . . 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 9585	. . 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 799	1 |- ( ph -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2160   _Vcvv 2752   C_ wss 3144   ifcif 3549   class class class wbr 4021   |-> cmpt 4082   |` cres 4649   ` cfv 5238   CCcc 7844   0cc0 7846   + caddc 7849   ZZcz 9288   ZZ>=cuz 9563   seqcseq 10484   ~~> cli 11327

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-addcom 7946  ax-addass 7948  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-0id 7954  ax-rnegex 7955  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-frec 6420  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-inn 8955  df-n0 9212  df-z 9289  df-uz 9564  df-fz 10045  df-fzo 10179  df-seqfrec 10485  df-clim 11328

This theorem is referenced by:  summodc  11432  zsumdc  11433