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Theorem fprodconst 28671
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 12126 . . . . 5  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
21eqcomd 2468 . . . 4  |-  ( B  e.  CC  ->  1  =  ( B ^
0 ) )
3 prodeq1 28604 . . . . . 6  |-  ( A  =  (/)  ->  prod_ k  e.  A  B  =  prod_ k  e.  (/)  B )
4 prod0 28638 . . . . . 6  |-  prod_ k  e.  (/)  B  =  1
53, 4syl6eq 2517 . . . . 5  |-  ( A  =  (/)  ->  prod_ k  e.  A  B  = 
1 )
6 fveq2 5857 . . . . . . 7  |-  ( A  =  (/)  ->  ( # `  A )  =  (
# `  (/) ) )
7 hash0 12392 . . . . . . 7  |-  ( # `  (/) )  =  0
86, 7syl6eq 2517 . . . . . 6  |-  ( A  =  (/)  ->  ( # `  A )  =  0 )
98oveq2d 6291 . . . . 5  |-  ( A  =  (/)  ->  ( B ^ ( # `  A
) )  =  ( B ^ 0 ) )
105, 9eqeq12d 2482 . . . 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 2461 . . . . . . 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 11703 . . . . . . . . 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 6105 . . . . . . . 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 28636 . . . . . 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 12125 . . . . . . 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 2504 . . . . 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 1695 . . 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 13481 . . 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 1374   E.wex 1591    e. wcel 1762   (/)c0 3778   {csn 4020    X. cxp 4990   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275   Fincfn 7506   CCcc 9479   0cc0 9481   1c1 9482    x. cmul 9486   NNcn 10525   ...cfz 11661    seqcseq 12063   ^cexp 12122   #chash 12360   prod_cprod 28600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-prod 28601
This theorem is referenced by:  risefallfac  28709
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