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Theorem prod1 14075
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 2471 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 simpr 468 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  M  e.  ZZ )
3 ax-1ne0 9626 . . . . 5  |-  1  =/=  0
43a1i 11 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  1  =/=  0 )
51prodfclim1 14026 . . . . 5  |-  ( M  e.  ZZ  ->  seq M (  x.  , 
( ( ZZ>= `  M
)  X.  { 1 } ) )  ~~>  1 )
65adantl 473 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  seq M (  x.  , 
( ( ZZ>= `  M
)  X.  { 1 } ) )  ~~>  1 )
7 simpl 464 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  A  C_  ( ZZ>= `  M )
)
8 1ex 9656 . . . . . . 7  |-  1  e.  _V
98fvconst2 6136 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 1 } ) `
 k )  =  1 )
10 ifid 3909 . . . . . 6  |-  if ( k  e.  A , 
1 ,  1 )  =  1
119, 10syl6eqr 2523 . . . . 5  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 1 } ) `
 k )  =  if ( k  e.  A ,  1 ,  1 ) )
1211adantl 473 . . . 4  |-  ( ( ( A  C_  ( ZZ>=
`  M )  /\  M  e.  ZZ )  /\  k  e.  ( ZZ>=
`  M ) )  ->  ( ( (
ZZ>= `  M )  X. 
{ 1 } ) `
 k )  =  if ( k  e.  A ,  1 ,  1 ) )
13 1cnd 9677 . . . 4  |-  ( ( ( A  C_  ( ZZ>=
`  M )  /\  M  e.  ZZ )  /\  k  e.  A
)  ->  1  e.  CC )
141, 2, 4, 6, 7, 12, 13zprodn0 14070 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  prod_ k  e.  A  1  =  1 )
15 uzf 11185 . . . . . . . . 9  |-  ZZ>= : ZZ --> ~P ZZ
1615fdmi 5746 . . . . . . . 8  |-  dom  ZZ>=  =  ZZ
1716eleq2i 2541 . . . . . . 7  |-  ( M  e.  dom  ZZ>=  <->  M  e.  ZZ )
18 ndmfv 5903 . . . . . . 7  |-  ( -.  M  e.  dom  ZZ>=  -> 
( ZZ>= `  M )  =  (/) )
1917, 18sylnbir 314 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  (
ZZ>= `  M )  =  (/) )
2019sseq2d 3446 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( A  C_  ( ZZ>= `  M )  <->  A  C_  (/) ) )
2120biimpac 494 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  A  C_  (/) )
22 ss0 3768 . . . 4  |-  ( A 
C_  (/)  ->  A  =  (/) )
23 prodeq1 14040 . . . . 5  |-  ( A  =  (/)  ->  prod_ k  e.  A  1  =  prod_ k  e.  (/)  1 )
24 prod0 14074 . . . . 5  |-  prod_ k  e.  (/)  1  =  1
2523, 24syl6eq 2521 . . . 4  |-  ( A  =  (/)  ->  prod_ k  e.  A  1  = 
1 )
2621, 22, 253syl 18 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  prod_ k  e.  A 
1  =  1 )
2714, 26pm2.61dan 808 . 2  |-  ( A 
C_  ( ZZ>= `  M
)  ->  prod_ k  e.  A  1  =  1 )
28 fz1f1o 13853 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
29 eqidd 2472 . . . . . . . . 9  |-  ( k  =  ( f `  j )  ->  1  =  1 )
30 simpl 464 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  ( # `
 A )  e.  NN )
31 simpr 468 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
32 1cnd 9677 . . . . . . . . 9  |-  ( ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  /\  k  e.  A )  ->  1  e.  CC )
33 elfznn 11854 . . . . . . . . . . 11  |-  ( j  e.  ( 1 ... ( # `  A
) )  ->  j  e.  NN )
348fvconst2 6136 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  (
( NN  X.  {
1 } ) `  j )  =  1 )
3533, 34syl 17 . . . . . . . . . 10  |-  ( j  e.  ( 1 ... ( # `  A
) )  ->  (
( NN  X.  {
1 } ) `  j )  =  1 )
3635adantl 473 . . . . . . . . 9  |-  ( ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  /\  j  e.  ( 1 ... ( # `
 A ) ) )  ->  ( ( NN  X.  { 1 } ) `  j )  =  1 )
3729, 30, 31, 32, 36fprod 14072 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  prod_ k  e.  A  1  =  (  seq 1 (  x.  ,  ( NN 
X.  { 1 } ) ) `  ( # `
 A ) ) )
38 nnuz 11218 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
3938prodf1 14024 . . . . . . . . 9  |-  ( (
# `  A )  e.  NN  ->  (  seq 1 (  x.  , 
( NN  X.  {
1 } ) ) `
 ( # `  A
) )  =  1 )
4039adantr 472 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  (  seq 1 (  x.  , 
( NN  X.  {
1 } ) ) `
 ( # `  A
) )  =  1 )
4137, 40eqtrd 2505 . . . . . . 7  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  prod_ k  e.  A  1  =  1 )
4241ex 441 . . . . . 6  |-  ( (
# `  A )  e.  NN  ->  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  prod_ k  e.  A  1  =  1 ) )
4342exlimdv 1787 . . . . 5  |-  ( (
# `  A )  e.  NN  ->  ( E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A  ->  prod_ k  e.  A  1  =  1 ) )
4443imp 436 . . . 4  |-  ( ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A )  ->  prod_ k  e.  A 
1  =  1 )
4525, 44jaoi 386 . . 3  |-  ( ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) )  ->  prod_ k  e.  A  1  =  1 )
4628, 45syl 17 . 2  |-  ( A  e.  Fin  ->  prod_ k  e.  A  1  =  1 )
4727, 46jaoi 386 1  |-  ( ( A  C_  ( ZZ>= `  M )  \/  A  e.  Fin )  ->  prod_ k  e.  A  1  =  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641    C_ wss 3390   (/)c0 3722   ifcif 3872   ~Pcpw 3942   {csn 3959   class class class wbr 4395    X. cxp 4837   dom cdm 4839   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   Fincfn 7587   0cc0 9557   1c1 9558    x. cmul 9562   NNcn 10631   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810    seqcseq 12251   #chash 12553    ~~> cli 13625   prod_cprod 14036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-prod 14037
This theorem is referenced by:  etransclem35  38246
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