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Theorem prodrblem 27288
Description: Lemma for prodrb 27291. (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 9371 . . 3  |-  ( n  e.  CC  ->  (
1  x.  n )  =  n )
21adantl 463 . 2  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  CC )  ->  ( 1  x.  n )  =  n )
3 ax-1cn 9327 . . 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 462 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  N  e.  (
ZZ>= `  M ) )
7 iftrue 3785 . . . . . . . . 9  |-  ( k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  B )
87adantl 463 . . . . . . . 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 707 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  e.  A )  ->  B  e.  CC )
118, 10eqeltrd 2507 . . . . . . 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 3787 . . . . . . 7  |-  ( -.  k  e.  A  ->  if ( k  e.  A ,  B ,  1 )  =  1 )
1413, 3syl6eqel 2521 . . . . . 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 5855 . . . 4  |-  ( ph  ->  F : ZZ --> CC )
18 uzssz 10867 . . . . 5  |-  ( ZZ>= `  M )  C_  ZZ
1918, 5sseldi 3342 . . . 4  |-  ( ph  ->  N  e.  ZZ )
2017, 19ffvelrnd 5832 . . 3  |-  ( ph  ->  ( F `  N
)  e.  CC )
2120adantr 462 . 2  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  ( F `  N )  e.  CC )
22 elfzelz 11439 . . . . 5  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  n  e.  ZZ )
2322adantl 463 . . . 4  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ZZ )
24 simplr 747 . . . . . 6  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A  C_  ( ZZ>=
`  N ) )
2519zcnd 10735 . . . . . . . . . 10  |-  ( ph  ->  N  e.  CC )
2625adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  A  C_  ( ZZ>=
`  N ) )  ->  N  e.  CC )
2726adantr 462 . . . . . . . 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 9710 . . . . . . 7  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ( N  -  1 )  +  1 )  =  N )
3029fveq2d 5683 . . . . . 6  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) )  =  (
ZZ>= `  N ) )
3124, 30sseqtr4d 3381 . . . . 5  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  A  C_  ( ZZ>=
`  ( ( N  -  1 )  +  1 ) ) )
32 fznuz 11525 . . . . . 6  |-  ( n  e.  ( M ... ( N  -  1
) )  ->  -.  n  e.  ( ZZ>= `  ( ( N  - 
1 )  +  1 ) ) )
3332adantl 463 . . . . 5  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -.  n  e.  ( ZZ>= `  ( ( N  -  1 )  +  1 ) ) )
3431, 33ssneldd 3347 . . . 4  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  -.  n  e.  A )
3523, 34eldifd 3327 . . 3  |-  ( ( ( ph  /\  A  C_  ( ZZ>= `  N )
)  /\  n  e.  ( M ... ( N  -  1 ) ) )  ->  n  e.  ( ZZ  \  A ) )
36 fveq2 5679 . . . . 5  |-  ( k  =  n  ->  ( F `  k )  =  ( F `  n ) )
3736eqeq1d 2441 . . . 4  |-  ( k  =  n  ->  (
( F `  k
)  =  1  <->  ( F `  n )  =  1 ) )
38 eldifi 3466 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  k  e.  ZZ )
39 eldifn 3467 . . . . . . . 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 2521 . . . . . 6  |-  ( k  e.  ( ZZ  \  A )  ->  if ( k  e.  A ,  B ,  1 )  e.  CC )
4216fvmpt2 5769 . . . . . 6  |-  ( ( k  e.  ZZ  /\  if ( k  e.  A ,  B ,  1 )  e.  CC )  -> 
( F `  k
)  =  if ( k  e.  A ,  B ,  1 ) )
4338, 41, 42syl2anc 654 . . . . 5  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  if ( k  e.  A ,  B , 
1 ) )
4443, 40eqtrd 2465 . . . 4  |-  ( k  e.  ( ZZ  \  A )  ->  ( F `  k )  =  1 )
4537, 44vtoclga 3025 . . 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 11834 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 1362    e. wcel 1755    \ cdif 3313    C_ wss 3316   ifcif 3779    e. cmpt 4338    |` cres 4829   ` cfv 5406  (class class class)co 6080   CCcc 9267   1c1 9270    + caddc 9272    x. cmul 9274    - cmin 9582   ZZcz 10633   ZZ>=cuz 10848   ...cfz 11423    seqcseq 11789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-nn 10310  df-n0 10567  df-z 10634  df-uz 10849  df-fz 11424  df-seq 11790
This theorem is referenced by:  prodrblem2  27290
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