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Theorem sumrblem 12184
Description: Lemma for sumrb 12186. (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 8995 . . 3  |-  ( n  e.  CC  ->  (
0  +  n )  =  n )
21adantl 452 . 2  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  CC )  ->  ( 0  +  n )  =  n )
3 0cn 8831 . . 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 3571 . . . . . . . . . 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 2357 . . . . . . . 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 3572 . . . . . . . 8  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
1312, 3syl6eqel 2371 . . . . . . 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 5684 . . . 4  |-  ( ph  ->  F : ZZ --> CC )
1817adantr 451 . . 3  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  F : ZZ --> CC )
19 eluzelz 10238 . . . . 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 5663 . . 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 10798 . . . . 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 10864 . . . . . 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 10118 . . . . . . . . . 10  |-  ( ph  ->  N  e.  CC )
3029ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  N  e.  CC )
31 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
32 npcan 9060 . . . . . . . . 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 5529 . . . . . . 7  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
3528, 34sseqtr4d 3215 . . . . . 6  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A  C_  ( ZZ>=
`  ( ( N  -  1 )  +  1 ) ) )
3635sseld 3179 . . . . 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 3162 . . . 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 5525 . . . . 5  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
4140eqeq1d 2291 . . . 4  |-  ( k  =  n  ->  (
( F `  k
)  =  0  <->  ( F `  n )  =  0 ) )
42 eldifi 3298 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
43 eldifn 3299 . . . . . . . 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 2371 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
4616fvmpt2 5608 . . . . . 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 2315 . . . 4  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  0 )
4941, 48vtoclga 2849 . . 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 11091 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 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   ifcif 3565    e. cmpt 4077    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    - cmin 9037   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046
This theorem is referenced by:  sumrb  12186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047
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