MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sumrblem Unicode version

Theorem sumrblem 12281
Description: Lemma for sumrb 12283. (Contributed by Mario Carneiro, 12-Aug-2013.)
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.3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
Assertion
Ref Expression
sumrblem  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  (  seq  M
(  +  ,  F
)  |`  ( ZZ>= `  N
) )  =  seq  N (  +  ,  F
) )
Distinct variable groups:    A, k    k, F    k, N    ph, k    k, M
Allowed substitution hint:    B( k)

Proof of Theorem sumrblem
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 addid2 9085 . . 3  |-  ( n  e.  CC  ->  (
0  +  n )  =  n )
21adantl 452 . 2  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  CC )  ->  ( 0  +  n )  =  n )
3 0cn 8921 . . 3  |-  0  e.  CC
43a1i 10 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  0  e.  CC )
5 sumrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
65adantr 451 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  N  e.  (
ZZ>= `  M ) )
7 iftrue 3647 . . . . . . . . . 10  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  B )
87adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  0 )  =  B )
9 summo.2 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
108, 9eqeltrd 2432 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
1110ex 423 . . . . . . 7  |-  ( ph  ->  ( k  e.  A  ->  if ( k  e.  A ,  B , 
0 )  e.  CC ) )
12 iffalse 3648 . . . . . . . 8  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
1312, 3syl6eqel 2446 . . . . . . 7  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
1411, 13pm2.61d1 151 . . . . . 6  |-  ( ph  ->  if ( k  e.  A ,  B , 
0 )  e.  CC )
1514adantr 451 . . . . 5  |-  ( (
ph  /\  k  e.  ZZ )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
16 summo.1 . . . . 5  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
1715, 16fmptd 5767 . . . 4  |-  ( ph  ->  F : ZZ --> CC )
1817adantr 451 . . 3  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  F : ZZ --> CC )
19 eluzelz 10330 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
205, 19syl 15 . . . 4  |-  ( ph  ->  N  e.  ZZ )
2120adantr 451 . . 3  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  N  e.  ZZ )
22 ffvelrn 5746 . . 3  |-  ( ( F : ZZ --> CC  /\  N  e.  ZZ )  ->  ( F `  N
)  e.  CC )
2318, 21, 22syl2anc 642 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  ( F `  N )  e.  CC )
24 elfzelz 10890 . . . . 5  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  n  e.  ZZ )
2524adantl 452 . . . 4  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ZZ )
26 fznuz 10956 . . . . . 6  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  -.  n  e.  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) ) )
2726adantl 452 . . . . 5  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -.  n  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
28 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A  C_  ( ZZ>=
`  N ) )
2920zcnd 10210 . . . . . . . . . 10  |-  ( ph  ->  N  e.  CC )
3029ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  N  e.  CC )
31 ax-1cn 8885 . . . . . . . . 9  |-  1  e.  CC
32 npcan 9150 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
3330, 31, 32sylancl 643 . . . . . . . 8  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ( N  -  1 )  +  1 )  =  N )
3433fveq2d 5612 . . . . . . 7  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
3528, 34sseqtr4d 3291 . . . . . 6  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A  C_  ( ZZ>=
`  ( ( N  -  1 )  +  1 ) ) )
3635sseld 3255 . . . . 5  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( n  e.  A  ->  n  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) ) )
3727, 36mtod 168 . . . 4  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -.  n  e.  A )
38 eldif 3238 . . . 4  |-  ( n  e.  ( ZZ  \  A )  <->  ( n  e.  ZZ  /\  -.  n  e.  A ) )
3925, 37, 38sylanbrc 645 . . 3  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ( ZZ  \  A ) )
40 fveq2 5608 . . . . 5  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
4140eqeq1d 2366 . . . 4  |-  ( k  =  n  ->  (
( F `  k
)  =  0  <->  ( F `  n )  =  0 ) )
42 eldifi 3374 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
43 eldifn 3375 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  -.  k  e.  A )
4443, 12syl 15 . . . . . . 7  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
4544, 3syl6eqel 2446 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
4616fvmpt2 5691 . . . . . 6  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  0 )  e.  CC )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  0 ) )
4742, 45, 46syl2anc 642 . . . . 5  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
0 ) )
4847, 44eqtrd 2390 . . . 4  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  0 )
4941, 48vtoclga 2925 . . 3  |-  ( n  e.  ( ZZ  \  A )  ->  ( F `  n )  =  0 )
5039, 49syl 15 . 2  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  n )  =  0 )
512, 4, 6, 23, 50seqid 11183 1  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  (  seq  M
(  +  ,  F
)  |`  ( ZZ>= `  N
) )  =  seq  N (  +  ,  F
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    \ cdif 3225    C_ wss 3228   ifcif 3641    e. cmpt 4158    |` cres 4773   -->wf 5333   ` cfv 5337  (class class class)co 5945   CCcc 8825   0cc0 8827   1c1 8828    + caddc 8830    - cmin 9127   ZZcz 10116   ZZ>=cuz 10322   ...cfz 10874    seq cseq 11138
This theorem is referenced by:  sumrb  12283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-seq 11139
  Copyright terms: Public domain W3C validator