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Theorem prodrblem 27609
Description: Lemma for prodrb 27612. (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 9499 . . 3  |-  ( n  e.  CC  ->  (
1  x.  n )  =  n )
21adantl 466 . 2  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  CC )  ->  ( 1  x.  n )  =  n )
3 ax-1cn 9455 . . 3  |-  1  e.  CC
43a1i 11 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  1  e.  CC )
5 prodrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
65adantr 465 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  N  e.  (
ZZ>= `  M ) )
7 iftrue 3908 . . . . . . . . 9  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  B )
87adantl 466 . . . . . . . 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 714 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  e.  A )  ->  B  e.  CC )
118, 10eqeltrd 2542 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
1211ex 434 . . . . . 6  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  e.  CC ) )
13 iffalse 3910 . . . . . . 7  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  1 )
1413, 3syl6eqel 2550 . . . . . 6  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
1512, 14pm2.61d1 159 . . . . 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 5979 . . . 4  |-  ( ph  ->  F : ZZ --> CC )
18 uzssz 10995 . . . . 5  |-  ( ZZ>= `  M )  C_  ZZ
1918, 5sseldi 3465 . . . 4  |-  ( ph  ->  N  e.  ZZ )
2017, 19ffvelrnd 5956 . . 3  |-  ( ph  ->  ( F `  N
)  e.  CC )
2120adantr 465 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  ( F `  N )  e.  CC )
22 elfzelz 11574 . . . . 5  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  n  e.  ZZ )
2322adantl 466 . . . 4  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ZZ )
24 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A  C_  ( ZZ>=
`  N ) )
2519zcnd 10863 . . . . . . . . . 10  |-  ( ph  ->  N  e.  CC )
2625adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  N  e.  CC )
2726adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  N  e.  CC )
283a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  1  e.  CC )
2927, 28npcand 9838 . . . . . . 7  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ( N  -  1 )  +  1 )  =  N )
3029fveq2d 5806 . . . . . 6  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
3124, 30sseqtr4d 3504 . . . . 5  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A  C_  ( ZZ>=
`  ( ( N  -  1 )  +  1 ) ) )
32 fznuz 11663 . . . . . 6  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  -.  n  e.  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) ) )
3332adantl 466 . . . . 5  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -.  n  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
3431, 33ssneldd 3470 . . . 4  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -.  n  e.  A )
3523, 34eldifd 3450 . . 3  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ( ZZ  \  A ) )
36 fveq2 5802 . . . . 5  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
3736eqeq1d 2456 . . . 4  |-  ( k  =  n  ->  (
( F `  k
)  =  1  <->  ( F `  n )  =  1 ) )
38 eldifi 3589 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
39 eldifn 3590 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  -.  k  e.  A )
4039, 13syl 16 . . . . . . 7  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  1 )  =  1 )
4140, 3syl6eqel 2550 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
4216fvmpt2 5893 . . . . . 6  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  1 )  e.  CC )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  1 ) )
4338, 41, 42syl2anc 661 . . . . 5  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
1 ) )
4443, 40eqtrd 2495 . . . 4  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  1 )
4537, 44vtoclga 3142 . . 3  |-  ( n  e.  ( ZZ  \  A )  ->  ( F `  n )  =  1 )
4635, 45syl 16 . 2  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  n )  =  1 )
472, 4, 6, 21, 46seqid 11972 1  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  (  seq M
(  x.  ,  F
)  |`  ( ZZ>= `  N
) )  =  seq N (  x.  ,  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    \ cdif 3436    C_ wss 3439   ifcif 3902    |-> cmpt 4461    |` cres 4953   ` cfv 5529  (class class class)co 6203   CCcc 9395   1c1 9398    + caddc 9400    x. cmul 9402    - cmin 9710   ZZcz 10761   ZZ>=cuz 10976   ...cfz 11558    seqcseq 11927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-seq 11928
This theorem is referenced by:  prodrblem2  27611
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