Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prod1 Unicode version

Theorem prod1 25223
Description: Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.)
Assertion
Ref Expression
prod1  |-  ( ( A  C_  ( ZZ>= `  M )  \/  A  e.  Fin )  ->  prod_ k  e.  A 1  =  1 )
Distinct variable groups:    A, k    k, M

Proof of Theorem prod1
Dummy variables  f 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 simpr 448 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  M  e.  ZZ )
3 ax-1ne0 9015 . . . . 5  |-  1  =/=  0
43a1i 11 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  1  =/=  0 )
51prodfclim1 25174 . . . . 5  |-  ( M  e.  ZZ  ->  seq  M (  x.  ,  ( ( ZZ>= `  M )  X.  { 1 } ) )  ~~>  1 )
65adantl 453 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  seq  M (  x.  ,  ( ( ZZ>= `  M )  X.  { 1 } ) )  ~~>  1 )
7 simpl 444 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  A  C_  ( ZZ>= `  M )
)
8 1ex 9042 . . . . . . 7  |-  1  e.  _V
98fvconst2 5906 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 1 } ) `
 k )  =  1 )
10 ifid 3731 . . . . . 6  |-  if ( k  e.  A , 
1 ,  1 )  =  1
119, 10syl6eqr 2454 . . . . 5  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 1 } ) `
 k )  =  if ( k  e.  A ,  1 ,  1 ) )
1211adantl 453 . . . 4  |-  ( ( ( A  C_  ( ZZ>=
`  M )  /\  M  e.  ZZ )  /\  k  e.  ( ZZ>=
`  M ) )  ->  ( ( (
ZZ>= `  M )  X. 
{ 1 } ) `
 k )  =  if ( k  e.  A ,  1 ,  1 ) )
13 ax-1cn 9004 . . . . 5  |-  1  e.  CC
1413a1i 11 . . . 4  |-  ( ( ( A  C_  ( ZZ>=
`  M )  /\  M  e.  ZZ )  /\  k  e.  A
)  ->  1  e.  CC )
151, 2, 4, 6, 7, 12, 14zprodn0 25218 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  prod_ k  e.  A 1  =  1 )
16 uzf 10447 . . . . . . . . 9  |-  ZZ>= : ZZ --> ~P ZZ
1716fdmi 5555 . . . . . . . 8  |-  dom  ZZ>=  =  ZZ
1817eleq2i 2468 . . . . . . 7  |-  ( M  e.  dom  ZZ>=  <->  M  e.  ZZ )
19 ndmfv 5714 . . . . . . 7  |-  ( -.  M  e.  dom  ZZ>=  -> 
( ZZ>= `  M )  =  (/) )
2018, 19sylnbir 299 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  (
ZZ>= `  M )  =  (/) )
2120sseq2d 3336 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( A  C_  ( ZZ>= `  M )  <->  A  C_  (/) ) )
2221biimpac 473 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  A  C_  (/) )
23 ss0 3618 . . . 4  |-  ( A 
C_  (/)  ->  A  =  (/) )
24 prodeq1 25188 . . . . 5  |-  ( A  =  (/)  ->  prod_ k  e.  A 1  =  prod_ k  e.  (/) 1 )
25 prod0 25222 . . . . 5  |-  prod_ k  e.  (/) 1  =  1
2624, 25syl6eq 2452 . . . 4  |-  ( A  =  (/)  ->  prod_ k  e.  A 1  =  1 )
2722, 23, 263syl 19 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  prod_ k  e.  A
1  =  1 )
2815, 27pm2.61dan 767 . 2  |-  ( A 
C_  ( ZZ>= `  M
)  ->  prod_ k  e.  A 1  =  1 )
29 fz1f1o 12459 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
30 eqidd 2405 . . . . . . . . 9  |-  ( k  =  ( f `  j )  ->  1  =  1 )
31 simpl 444 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  ( # `
 A )  e.  NN )
32 simpr 448 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
3313a1i 11 . . . . . . . . 9  |-  ( ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  /\  k  e.  A )  ->  1  e.  CC )
34 elfznn 11036 . . . . . . . . . . 11  |-  ( j  e.  ( 1 ... ( # `  A
) )  ->  j  e.  NN )
358fvconst2 5906 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  (
( NN  X.  {
1 } ) `  j )  =  1 )
3634, 35syl 16 . . . . . . . . . 10  |-  ( j  e.  ( 1 ... ( # `  A
) )  ->  (
( NN  X.  {
1 } ) `  j )  =  1 )
3736adantl 453 . . . . . . . . 9  |-  ( ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  /\  j  e.  ( 1 ... ( # `
 A ) ) )  ->  ( ( NN  X.  { 1 } ) `  j )  =  1 )
3830, 31, 32, 33, 37fprod 25220 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  prod_ k  e.  A 1  =  (  seq  1 (  x.  ,  ( NN 
X.  { 1 } ) ) `  ( # `
 A ) ) )
39 nnuz 10477 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
4039prodf1 25172 . . . . . . . . 9  |-  ( (
# `  A )  e.  NN  ->  (  seq  1 (  x.  , 
( NN  X.  {
1 } ) ) `
 ( # `  A
) )  =  1 )
4140adantr 452 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  (  seq  1 (  x.  , 
( NN  X.  {
1 } ) ) `
 ( # `  A
) )  =  1 )
4238, 41eqtrd 2436 . . . . . . 7  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  prod_ k  e.  A 1  =  1 )
4342ex 424 . . . . . 6  |-  ( (
# `  A )  e.  NN  ->  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  prod_ k  e.  A 1  =  1 ) )
4443exlimdv 1643 . . . . 5  |-  ( (
# `  A )  e.  NN  ->  ( E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A  ->  prod_ k  e.  A 1  =  1 ) )
4544imp 419 . . . 4  |-  ( ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A )  ->  prod_ k  e.  A
1  =  1 )
4626, 45jaoi 369 . . 3  |-  ( ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) )  ->  prod_ k  e.  A 1  =  1 )
4729, 46syl 16 . 2  |-  ( A  e.  Fin  ->  prod_ k  e.  A 1  =  1 )
4828, 47jaoi 369 1  |-  ( ( A  C_  ( ZZ>= `  M )  \/  A  e.  Fin )  ->  prod_ k  e.  A 1  =  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567    C_ wss 3280   (/)c0 3588   ifcif 3699   ~Pcpw 3759   {csn 3774   class class class wbr 4172    X. cxp 4835   dom cdm 4837   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   0cc0 8946   1c1 8947    x. cmul 8951   NNcn 9956   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278   #chash 11573    ~~> cli 12233   prod_cprod 25184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-prod 25185
  Copyright terms: Public domain W3C validator