Theorem sumrb 14444

Description: Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 9-Apr-2014.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem sumrb

Step	Hyp	Ref	Expression
1		sumrb.5	. . . . 5 |- ( ph -> N e. ZZ )
2	1	adantr 481	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> N e. ZZ )
3		seqex 12803	. . . 4 |- seq M ( + , F ) e. _V
4		climres 14306	. . . 4 |- ( ( N e. ZZ /\ seq M ( + , F ) e. _V ) -> ( ( seq M ( + , F ) |` ( ZZ>= ` N ) ) ~~> C <-> seq M ( + , F ) ~~> C ) )
5	2, 3, 4	sylancl 694	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( ( seq M ( + , F ) |` ( ZZ>= ` N ) ) ~~> C <-> seq M ( + , F ) ~~> C ) )
6		sumrb.7	. . . . 5 |- ( ph -> A C_ ( ZZ>= ` N ) )
7		summo.1	. . . . . 6 |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
8		summo.2	. . . . . . 7 |- ( ( ph /\ k e. A ) -> B e. CC )
9	8	adantlr 751	. . . . . 6 |- ( ( ( ph /\ N e. ( ZZ>= ` M ) ) /\ k e. A ) -> B e. CC )
10		simpr 477	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> N e. ( ZZ>= ` M ) )
11	7, 9, 10	sumrblem 14442	. . . . 5 |- ( ( ( ph /\ N e. ( ZZ>= ` M ) ) /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )
12	6, 11	mpidan 704	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )
13	12	breq1d 4663	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( ( seq M ( + , F ) |` ( ZZ>= ` N ) ) ~~> C <-> seq N ( + , F ) ~~> C ) )
14	5, 13	bitr3d 270	. 2 |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )
15		sumrb.6	. . . . 5 |- ( ph -> A C_ ( ZZ>= ` M ) )
16	8	adantlr 751	. . . . . 6 |- ( ( ( ph /\ M e. ( ZZ>= ` N ) ) /\ k e. A ) -> B e. CC )
17		simpr 477	. . . . . 6 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> M e. ( ZZ>= ` N ) )
18	7, 16, 17	sumrblem 14442	. . . . 5 |- ( ( ( ph /\ M e. ( ZZ>= ` N ) ) /\ A C_ ( ZZ>= ` M ) ) -> ( seq N ( + , F ) |` ( ZZ>= ` M ) ) = seq M ( + , F ) )
19	15, 18	mpidan 704	. . . 4 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> ( seq N ( + , F ) |` ( ZZ>= ` M ) ) = seq M ( + , F ) )
20	19	breq1d 4663	. . 3 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> ( ( seq N ( + , F ) |` ( ZZ>= ` M ) ) ~~> C <-> seq M ( + , F ) ~~> C ) )
21		sumrb.4	. . . . 5 |- ( ph -> M e. ZZ )
22	21	adantr 481	. . . 4 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> M e. ZZ )
23		seqex 12803	. . . 4 |- seq N ( + , F ) e. _V
24		climres 14306	. . . 4 |- ( ( M e. ZZ /\ seq N ( + , F ) e. _V ) -> ( ( seq N ( + , F ) |` ( ZZ>= ` M ) ) ~~> C <-> seq N ( + , F ) ~~> C ) )
25	22, 23, 24	sylancl 694	. . 3 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> ( ( seq N ( + , F ) |` ( ZZ>= ` M ) ) ~~> C <-> seq N ( + , F ) ~~> C ) )
26	20, 25	bitr3d 270	. 2 |- ( ( ph /\ M e. ( ZZ>= ` N ) ) -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )
27		uztric 11709	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) )
28	21, 1, 27	syl2anc 693	. 2 |- ( ph -> ( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) )
29	14, 26, 28	mpjaodan 827	1 |- ( ph -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   |` cres 5116   ` cfv 5888   CCcc 9934   0cc0 9936   + caddc 9939   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-clim 14219

This theorem is referenced by:  summo  14448  zsum  14449