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Theorem fprodconst 27458
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 11861 . . . . 5  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
21eqcomd 2443 . . . 4  |-  ( B  e.  CC  ->  1  =  ( B ^
0 ) )
3 prodeq1 27391 . . . . . 6  |-  ( A  =  (/)  ->  prod_ k  e.  A  B  =  prod_ k  e.  (/)  B )
4 prod0 27425 . . . . . 6  |-  prod_ k  e.  (/)  B  =  1
53, 4syl6eq 2486 . . . . 5  |-  ( A  =  (/)  ->  prod_ k  e.  A  B  = 
1 )
6 fveq2 5686 . . . . . . 7  |-  ( A  =  (/)  ->  ( # `  A )  =  (
# `  (/) ) )
7 hash0 12127 . . . . . . 7  |-  ( # `  (/) )  =  0
86, 7syl6eq 2486 . . . . . 6  |-  ( A  =  (/)  ->  ( # `  A )  =  0 )
98oveq2d 6102 . . . . 5  |-  ( A  =  (/)  ->  ( B ^ ( # `  A
) )  =  ( B ^ 0 ) )
105, 9eqeq12d 2452 . . . 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 466 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( A  =  (/)  ->  prod_ k  e.  A  B  =  ( B ^ ( # `  A
) ) ) )
13 eqidd 2439 . . . . . . 7  |-  ( k  =  ( f `  n )  ->  B  =  B )
14 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  -> 
( # `  A )  e.  NN )
15 simprr 756 . . . . . . 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 758 . . . . . . 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 758 . . . . . . . 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 11470 . . . . . . . . 9  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  n  e.  NN )
1918adantl 466 . . . . . . . 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 5926 . . . . . . . 8  |-  ( ( B  e.  CC  /\  n  e.  NN )  ->  ( ( NN  X.  { B } ) `  n )  =  B )
2117, 19, 20syl2anc 661 . . . . . . 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 27423 . . . . . 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 11860 . . . . . . 7  |-  ( ( B  e.  CC  /\  ( # `  A )  e.  NN )  -> 
( B ^ ( # `
 A ) )  =  (  seq 1
(  x.  ,  ( NN  X.  { B } ) ) `  ( # `  A ) ) )
2423ad2ant2lr 747 . . . . . 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 2473 . . . . 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 615 . . . 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 1690 . . 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 603 . 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 13179 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
3029adantr 465 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( A  =  (/)  \/  ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
3112, 28, 30mpjaod 381 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 368    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   (/)c0 3632   {csn 3872    X. cxp 4833   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   Fincfn 7302   CCcc 9272   0cc0 9274   1c1 9275    x. cmul 9279   NNcn 10314   ...cfz 11429    seqcseq 11798   ^cexp 11857   #chash 12095   prod_cprod 27387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-prod 27388
This theorem is referenced by:  risefallfac  27496
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