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Theorem sumrblem 12510
Description: Lemma for sumrb 12512. (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 9254 . . 3  |-  ( n  e.  CC  ->  (
0  +  n )  =  n )
21adantl 454 . 2  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  CC )  ->  ( 0  +  n )  =  n )
3 0cn 9089 . . 3  |-  0  e.  CC
43a1i 11 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  0  e.  CC )
5 sumrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
65adantr 453 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  N  e.  (
ZZ>= `  M ) )
7 iftrue 3747 . . . . . . . . . 10  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  B )
87adantl 454 . . . . . . . . 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 2512 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
1110ex 425 . . . . . . 7  |-  ( ph  ->  ( k  e.  A  ->  if ( k  e.  A ,  B , 
0 )  e.  CC ) )
12 iffalse 3748 . . . . . . . 8  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
1312, 3syl6eqel 2526 . . . . . . 7  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
1411, 13pm2.61d1 154 . . . . . 6  |-  ( ph  ->  if ( k  e.  A ,  B , 
0 )  e.  CC )
1514adantr 453 . . . . 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 5896 . . . 4  |-  ( ph  ->  F : ZZ --> CC )
1817adantr 453 . . 3  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  F : ZZ --> CC )
19 eluzelz 10501 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
205, 19syl 16 . . . 4  |-  ( ph  ->  N  e.  ZZ )
2120adantr 453 . . 3  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  N  e.  ZZ )
2218, 21ffvelrnd 5874 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  ( F `  N )  e.  CC )
23 elfzelz 11064 . . . . 5  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  n  e.  ZZ )
2423adantl 454 . . . 4  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ZZ )
25 simplr 733 . . . . . 6  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A  C_  ( ZZ>=
`  N ) )
2620zcnd 10381 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
2726ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  N  e.  CC )
28 ax-1cn 9053 . . . . . . . 8  |-  1  e.  CC
29 npcan 9319 . . . . . . . 8  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
3027, 28, 29sylancl 645 . . . . . . 7  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ( N  -  1 )  +  1 )  =  N )
3130fveq2d 5735 . . . . . 6  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
3225, 31sseqtr4d 3387 . . . . 5  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A  C_  ( ZZ>=
`  ( ( N  -  1 )  +  1 ) ) )
33 fznuz 11134 . . . . . 6  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  -.  n  e.  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) ) )
3433adantl 454 . . . . 5  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -.  n  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
3532, 34ssneldd 3353 . . . 4  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -.  n  e.  A )
3624, 35eldifd 3333 . . 3  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ( ZZ  \  A ) )
37 fveq2 5731 . . . . 5  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
3837eqeq1d 2446 . . . 4  |-  ( k  =  n  ->  (
( F `  k
)  =  0  <->  ( F `  n )  =  0 ) )
39 eldifi 3471 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
40 eldifn 3472 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  -.  k  e.  A )
4140, 12syl 16 . . . . . . 7  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  0 )  =  0 )
4241, 3syl6eqel 2526 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  0 )  e.  CC )
4316fvmpt2 5815 . . . . . 6  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  0 )  e.  CC )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  0 ) )
4439, 42, 43syl2anc 644 . . . . 5  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
0 ) )
4544, 41eqtrd 2470 . . . 4  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  0 )
4638, 45vtoclga 3019 . . 3  |-  ( n  e.  ( ZZ  \  A )  ->  ( F `  n )  =  0 )
4736, 46syl 16 . 2  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  n )  =  0 )
482, 4, 6, 22, 47seqid 11373 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 360    = wceq 1653    e. wcel 1726    \ cdif 3319    C_ wss 3322   ifcif 3741    e. cmpt 4269    |` cres 4883   -->wf 5453   ` cfv 5457  (class class class)co 6084   CCcc 8993   0cc0 8995   1c1 8996    + caddc 8998    - cmin 9296   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048    seq cseq 11328
This theorem is referenced by:  sumrb  12512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-seq 11329
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