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Theorem prodrblem 24432
Description: Lemma for prodrb 24435. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
prodmo.1  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
prodmo.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
prodrb.3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
Assertion
Ref Expression
prodrblem  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  (  seq  M
(  x.  ,  F
)  |`  ( ZZ>= `  N
) )  =  seq  N (  x.  ,  F
) )
Distinct variable groups:    A, k    k, F    ph, k
Allowed substitution hints:    B( k)    M( k)    N( k)

Proof of Theorem prodrblem
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 mulid2 8881 . . 3  |-  ( n  e.  CC  ->  (
1  x.  n )  =  n )
21adantl 452 . 2  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  CC )  ->  ( 1  x.  n )  =  n )
3 ax-1cn 8840 . . 3  |-  1  e.  CC
43a1i 10 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  1  e.  CC )
5 prodrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
65adantr 451 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  N  e.  (
ZZ>= `  M ) )
7 iftrue 3605 . . . . . . . . 9  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  B )
87adantl 452 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  1 )  =  B )
9 prodmo.2 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
109adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  e.  A )  ->  B  e.  CC )
118, 10eqeltrd 2390 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
1211ex 423 . . . . . 6  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  e.  CC ) )
13 iffalse 3606 . . . . . . 7  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  1 )
1413, 3syl6eqel 2404 . . . . . 6  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
1512, 14pm2.61d1 151 . . . . 5  |-  ( (
ph  /\  k  e.  ZZ )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
16 prodmo.1 . . . . 5  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
1715, 16fmptd 5722 . . . 4  |-  ( ph  ->  F : ZZ --> CC )
18 uzssz 10294 . . . . 5  |-  ( ZZ>= `  M )  C_  ZZ
1918, 5sseldi 3212 . . . 4  |-  ( ph  ->  N  e.  ZZ )
2017, 19ffvelrnd 5704 . . 3  |-  ( ph  ->  ( F `  N
)  e.  CC )
2120adantr 451 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  ( F `  N )  e.  CC )
22 elfzelz 10845 . . . . 5  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  n  e.  ZZ )
2322adantl 452 . . . 4  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ZZ )
24 simplr 731 . . . . . 6  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A  C_  ( ZZ>=
`  N ) )
2519zcnd 10165 . . . . . . . . . 10  |-  ( ph  ->  N  e.  CC )
2625adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  N  e.  CC )
2726adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  N  e.  CC )
283a1i 10 . . . . . . . 8  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  1  e.  CC )
2927, 28npcand 9206 . . . . . . 7  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ( N  -  1 )  +  1 )  =  N )
3029fveq2d 5567 . . . . . 6  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
3124, 30sseqtr4d 3249 . . . . 5  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A  C_  ( ZZ>=
`  ( ( N  -  1 )  +  1 ) ) )
32 fznuz 10911 . . . . . 6  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  -.  n  e.  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) ) )
3332adantl 452 . . . . 5  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -.  n  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
3431, 33ssneldd 3217 . . . 4  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -.  n  e.  A )
3523, 34eldifd 3197 . . 3  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ( ZZ  \  A ) )
36 fveq2 5563 . . . . 5  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
3736eqeq1d 2324 . . . 4  |-  ( k  =  n  ->  (
( F `  k
)  =  1  <->  ( F `  n )  =  1 ) )
38 eldifi 3332 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
39 eldifn 3333 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  -.  k  e.  A )
4039, 13syl 15 . . . . . . 7  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  1 )  =  1 )
4140, 3syl6eqel 2404 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
4216fvmpt2 5646 . . . . . 6  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  1 )  e.  CC )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  1 ) )
4338, 41, 42syl2anc 642 . . . . 5  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
1 ) )
4443, 40eqtrd 2348 . . . 4  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  1 )
4537, 44vtoclga 2883 . . 3  |-  ( n  e.  ( ZZ  \  A )  ->  ( F `  n )  =  1 )
4635, 45syl 15 . 2  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  n )  =  1 )
472, 4, 6, 21, 46seqid 11138 1  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  (  seq  M
(  x.  ,  F
)  |`  ( ZZ>= `  N
) )  =  seq  N (  x.  ,  F
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701    \ cdif 3183    C_ wss 3186   ifcif 3599    e. cmpt 4114    |` cres 4728   ` cfv 5292  (class class class)co 5900   CCcc 8780   1c1 8783    + caddc 8785    x. cmul 8787    - cmin 9082   ZZcz 10071   ZZ>=cuz 10277   ...cfz 10829    seq cseq 11093
This theorem is referenced by:  prodrblem2  24434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-seq 11094
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