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Theorem fprodconst 13934
Description: The product of constant terms ( k is not free in  B.) (Contributed by Scott Fenton, 12-Jan-2018.)
Assertion
Ref Expression
fprodconst  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  prod_ k  e.  A  B  =  ( B ^ ( # `  A
) ) )
Distinct variable groups:    A, k    B, k

Proof of Theorem fprodconst
Dummy variables  f  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exp0 12214 . . . . 5  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
21eqcomd 2410 . . . 4  |-  ( B  e.  CC  ->  1  =  ( B ^
0 ) )
3 prodeq1 13868 . . . . . 6  |-  ( A  =  (/)  ->  prod_ k  e.  A  B  =  prod_ k  e.  (/)  B )
4 prod0 13902 . . . . . 6  |-  prod_ k  e.  (/)  B  =  1
53, 4syl6eq 2459 . . . . 5  |-  ( A  =  (/)  ->  prod_ k  e.  A  B  = 
1 )
6 fveq2 5849 . . . . . . 7  |-  ( A  =  (/)  ->  ( # `  A )  =  (
# `  (/) ) )
7 hash0 12485 . . . . . . 7  |-  ( # `  (/) )  =  0
86, 7syl6eq 2459 . . . . . 6  |-  ( A  =  (/)  ->  ( # `  A )  =  0 )
98oveq2d 6294 . . . . 5  |-  ( A  =  (/)  ->  ( B ^ ( # `  A
) )  =  ( B ^ 0 ) )
105, 9eqeq12d 2424 . . . 4  |-  ( A  =  (/)  ->  ( prod_
k  e.  A  B  =  ( B ^
( # `  A ) )  <->  1  =  ( B ^ 0 ) ) )
112, 10syl5ibrcom 222 . . 3  |-  ( B  e.  CC  ->  ( A  =  (/)  ->  prod_ k  e.  A  B  =  ( B ^ ( # `
 A ) ) ) )
1211adantl 464 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( A  =  (/)  ->  prod_ k  e.  A  B  =  ( B ^ ( # `  A
) ) ) )
13 eqidd 2403 . . . . . . 7  |-  ( k  =  ( f `  n )  ->  B  =  B )
14 simprl 756 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  -> 
( # `  A )  e.  NN )
15 simprr 758 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  -> 
f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
16 simpllr 761 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  CC )  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  k  e.  A )  ->  B  e.  CC )
17 simpllr 761 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  CC )  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  n  e.  ( 1 ... ( # `  A
) ) )  ->  B  e.  CC )
18 elfznn 11768 . . . . . . . . 9  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  n  e.  NN )
1918adantl 464 . . . . . . . 8  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  CC )  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  n  e.  ( 1 ... ( # `  A
) ) )  ->  n  e.  NN )
20 fvconst2g 6105 . . . . . . . 8  |-  ( ( B  e.  CC  /\  n  e.  NN )  ->  ( ( NN  X.  { B } ) `  n )  =  B )
2117, 19, 20syl2anc 659 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  CC )  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  n  e.  ( 1 ... ( # `  A
) ) )  -> 
( ( NN  X.  { B } ) `  n )  =  B )
2213, 14, 15, 16, 21fprod 13900 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  ->  prod_ k  e.  A  B  =  (  seq 1
(  x.  ,  ( NN  X.  { B } ) ) `  ( # `  A ) ) )
23 expnnval 12213 . . . . . . 7  |-  ( ( B  e.  CC  /\  ( # `  A )  e.  NN )  -> 
( B ^ ( # `
 A ) )  =  (  seq 1
(  x.  ,  ( NN  X.  { B } ) ) `  ( # `  A ) ) )
2423ad2ant2lr 746 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  -> 
( B ^ ( # `
 A ) )  =  (  seq 1
(  x.  ,  ( NN  X.  { B } ) ) `  ( # `  A ) ) )
2522, 24eqtr4d 2446 . . . . 5  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  ->  prod_ k  e.  A  B  =  ( B ^
( # `  A ) ) )
2625expr 613 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( # `  A
)  e.  NN )  ->  ( f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A  ->  prod_ k  e.  A  B  =  ( B ^ ( # `  A
) ) ) )
2726exlimdv 1745 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( # `  A
)  e.  NN )  ->  ( E. f 
f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  prod_ k  e.  A  B  =  ( B ^ ( # `  A ) ) ) )
2827expimpd 601 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( ( ( # `  A )  e.  NN  /\ 
E. f  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A )  ->  prod_ k  e.  A  B  =  ( B ^ ( # `  A ) ) ) )
29 fz1f1o 13681 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
3029adantr 463 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( A  =  (/)  \/  ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
3112, 28, 30mpjaod 379 1  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  prod_ k  e.  A  B  =  ( B ^ ( # `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   (/)c0 3738   {csn 3972    X. cxp 4821   -1-1-onto->wf1o 5568   ` cfv 5569  (class class class)co 6278   Fincfn 7554   CCcc 9520   0cc0 9522   1c1 9523    x. cmul 9527   NNcn 10576   ...cfz 11726    seqcseq 12151   ^cexp 12210   #chash 12452   prod_cprod 13864
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-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  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  ax-pre-sup 9600
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-rmo 2762  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-int 4228  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-se 4783  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-isom 5578  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-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-prod 13865
This theorem is referenced by:  risefallfac  13969  bcprod  29947  etransclem23  37408
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