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Theorem prod1 25262
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 2435 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 simpr 448 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  M  e.  ZZ )
3 ax-1ne0 9051 . . . . 5  |-  1  =/=  0
43a1i 11 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  1  =/=  0 )
51prodfclim1 25213 . . . . 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 9078 . . . . . . 7  |-  1  e.  _V
98fvconst2 5939 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 1 } ) `
 k )  =  1 )
10 ifid 3763 . . . . . 6  |-  if ( k  e.  A , 
1 ,  1 )  =  1
119, 10syl6eqr 2485 . . . . 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 9040 . . . . 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 25257 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  prod_ k  e.  A 1  =  1 )
16 uzf 10483 . . . . . . . . 9  |-  ZZ>= : ZZ --> ~P ZZ
1716fdmi 5588 . . . . . . . 8  |-  dom  ZZ>=  =  ZZ
1817eleq2i 2499 . . . . . . 7  |-  ( M  e.  dom  ZZ>=  <->  M  e.  ZZ )
19 ndmfv 5747 . . . . . . 7  |-  ( -.  M  e.  dom  ZZ>=  -> 
( ZZ>= `  M )  =  (/) )
2018, 19sylnbir 299 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  (
ZZ>= `  M )  =  (/) )
2120sseq2d 3368 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( A  C_  ( ZZ>= `  M )  <->  A  C_  (/) ) )
2221biimpac 473 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  A  C_  (/) )
23 ss0 3650 . . . 4  |-  ( A 
C_  (/)  ->  A  =  (/) )
24 prodeq1 25227 . . . . 5  |-  ( A  =  (/)  ->  prod_ k  e.  A 1  =  prod_ k  e.  (/) 1 )
25 prod0 25261 . . . . 5  |-  prod_ k  e.  (/) 1  =  1
2624, 25syl6eq 2483 . . . 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 12496 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
30 eqidd 2436 . . . . . . . . 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 11072 . . . . . . . . . . 11  |-  ( j  e.  ( 1 ... ( # `  A
) )  ->  j  e.  NN )
358fvconst2 5939 . . . . . . . . . . 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 25259 . . . . . . . 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 10513 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
4039prodf1 25211 . . . . . . . . 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 2467 . . . . . . 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 1646 . . . . 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 1550    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   ifcif 3731   ~Pcpw 3791   {csn 3806   class class class wbr 4204    X. cxp 4868   dom cdm 4870   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   Fincfn 7101   CCcc 8980   0cc0 8982   1c1 8983    x. cmul 8987   NNcn 9992   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035    seq cseq 11315   #chash 11610    ~~> cli 12270   prod_cprod 25223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-prod 25224
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