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Theorem prodrblem 13888
Description: Lemma for prodrb 13891. (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 9624 . . 3  |-  ( n  e.  CC  ->  (
1  x.  n )  =  n )
21adantl 464 . 2  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  CC )  ->  ( 1  x.  n )  =  n )
3 1cnd 9642 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  1  e.  CC )
4 prodrb.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
54adantr 463 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  N  e.  (
ZZ>= `  M ) )
6 iftrue 3891 . . . . . . . . 9  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  B )
76adantl 464 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  1 )  =  B )
8 prodmo.2 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
98adantlr 713 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  e.  A )  ->  B  e.  CC )
107, 9eqeltrd 2490 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  e.  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
1110ex 432 . . . . . 6  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  e.  CC ) )
12 iffalse 3894 . . . . . . 7  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  1 )
13 ax-1cn 9580 . . . . . . 7  |-  1  e.  CC
1412, 13syl6eqel 2498 . . . . . 6  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
1511, 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 6033 . . . 4  |-  ( ph  ->  F : ZZ --> CC )
18 uzssz 11146 . . . . 5  |-  ( ZZ>= `  M )  C_  ZZ
1918, 4sseldi 3440 . . . 4  |-  ( ph  ->  N  e.  ZZ )
2017, 19ffvelrnd 6010 . . 3  |-  ( ph  ->  ( F `  N
)  e.  CC )
2120adantr 463 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  ( F `  N )  e.  CC )
22 elfzelz 11742 . . . . 5  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  n  e.  ZZ )
2322adantl 464 . . . 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 11009 . . . . . . . . . 10  |-  ( ph  ->  N  e.  CC )
2625adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  N  e.  CC )
2726adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  N  e.  CC )
28 1cnd 9642 . . . . . . . 8  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  1  e.  CC )
2927, 28npcand 9971 . . . . . . 7  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ( N  -  1 )  +  1 )  =  N )
3029fveq2d 5853 . . . . . 6  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
3124, 30sseqtr4d 3479 . . . . 5  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A  C_  ( ZZ>=
`  ( ( N  -  1 )  +  1 ) ) )
32 fznuz 11815 . . . . . 6  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  -.  n  e.  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) ) )
3332adantl 464 . . . . 5  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -.  n  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
3431, 33ssneldd 3445 . . . 4  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -.  n  e.  A )
3523, 34eldifd 3425 . . 3  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ( ZZ  \  A ) )
36 fveq2 5849 . . . . 5  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
3736eqeq1d 2404 . . . 4  |-  ( k  =  n  ->  (
( F `  k
)  =  1  <->  ( F `  n )  =  1 ) )
38 eldifi 3565 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
39 eldifn 3566 . . . . . . . 8  |-  ( k  e.  ( ZZ  \  A )  ->  -.  k  e.  A )
4039, 12syl 17 . . . . . . 7  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  1 )  =  1 )
4140, 13syl6eqel 2498 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
4216fvmpt2 5941 . . . . . 6  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  1 )  e.  CC )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  1 ) )
4338, 41, 42syl2anc 659 . . . . 5  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
1 ) )
4443, 40eqtrd 2443 . . . 4  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  1 )
4537, 44vtoclga 3123 . . 3  |-  ( n  e.  ( ZZ  \  A )  ->  ( F `  n )  =  1 )
4635, 45syl 17 . 2  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( F `  n )  =  1 )
472, 3, 5, 21, 46seqid 12196 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 367    = wceq 1405    e. wcel 1842    \ cdif 3411    C_ wss 3414   ifcif 3885    |-> cmpt 4453    |` cres 4825   ` cfv 5569  (class class class)co 6278   CCcc 9520   1c1 9523    + caddc 9525    x. cmul 9527    - cmin 9841   ZZcz 10905   ZZ>=cuz 11127   ...cfz 11726    seqcseq 12151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-seq 12152
This theorem is referenced by:  prodrblem2  13890
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