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Theorem prod1 13753
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 2382 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 simpr 459 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  M  e.  ZZ )
3 ax-1ne0 9472 . . . . 5  |-  1  =/=  0
43a1i 11 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  1  =/=  0 )
51prodfclim1 13704 . . . . 5  |-  ( M  e.  ZZ  ->  seq M (  x.  , 
( ( ZZ>= `  M
)  X.  { 1 } ) )  ~~>  1 )
65adantl 464 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  seq M (  x.  , 
( ( ZZ>= `  M
)  X.  { 1 } ) )  ~~>  1 )
7 simpl 455 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  A  C_  ( ZZ>= `  M )
)
8 1ex 9502 . . . . . . 7  |-  1  e.  _V
98fvconst2 6029 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 1 } ) `
 k )  =  1 )
10 ifid 3894 . . . . . 6  |-  if ( k  e.  A , 
1 ,  1 )  =  1
119, 10syl6eqr 2441 . . . . 5  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 1 } ) `
 k )  =  if ( k  e.  A ,  1 ,  1 ) )
1211adantl 464 . . . 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 9523 . . . 4  |-  ( ( ( A  C_  ( ZZ>=
`  M )  /\  M  e.  ZZ )  /\  k  e.  A
)  ->  1  e.  CC )
141, 2, 4, 6, 7, 12, 13zprodn0 13748 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  prod_ k  e.  A  1  =  1 )
15 uzf 11004 . . . . . . . . 9  |-  ZZ>= : ZZ --> ~P ZZ
1615fdmi 5644 . . . . . . . 8  |-  dom  ZZ>=  =  ZZ
1716eleq2i 2460 . . . . . . 7  |-  ( M  e.  dom  ZZ>=  <->  M  e.  ZZ )
18 ndmfv 5798 . . . . . . 7  |-  ( -.  M  e.  dom  ZZ>=  -> 
( ZZ>= `  M )  =  (/) )
1917, 18sylnbir 305 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  (
ZZ>= `  M )  =  (/) )
2019sseq2d 3445 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( A  C_  ( ZZ>= `  M )  <->  A  C_  (/) ) )
2120biimpac 484 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  A  C_  (/) )
22 ss0 3743 . . . 4  |-  ( A 
C_  (/)  ->  A  =  (/) )
23 prodeq1 13718 . . . . 5  |-  ( A  =  (/)  ->  prod_ k  e.  A  1  =  prod_ k  e.  (/)  1 )
24 prod0 13752 . . . . 5  |-  prod_ k  e.  (/)  1  =  1
2523, 24syl6eq 2439 . . . 4  |-  ( A  =  (/)  ->  prod_ k  e.  A  1  = 
1 )
2621, 22, 253syl 20 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  prod_ k  e.  A 
1  =  1 )
2714, 26pm2.61dan 789 . 2  |-  ( A 
C_  ( ZZ>= `  M
)  ->  prod_ k  e.  A  1  =  1 )
28 fz1f1o 13534 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
29 eqidd 2383 . . . . . . . . 9  |-  ( k  =  ( f `  j )  ->  1  =  1 )
30 simpl 455 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  ( # `
 A )  e.  NN )
31 simpr 459 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
32 1cnd 9523 . . . . . . . . 9  |-  ( ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  /\  k  e.  A )  ->  1  e.  CC )
33 elfznn 11635 . . . . . . . . . . 11  |-  ( j  e.  ( 1 ... ( # `  A
) )  ->  j  e.  NN )
348fvconst2 6029 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  (
( NN  X.  {
1 } ) `  j )  =  1 )
3533, 34syl 16 . . . . . . . . . 10  |-  ( j  e.  ( 1 ... ( # `  A
) )  ->  (
( NN  X.  {
1 } ) `  j )  =  1 )
3635adantl 464 . . . . . . . . 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 13750 . . . . . . . 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 11036 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
3938prodf1 13702 . . . . . . . . 9  |-  ( (
# `  A )  e.  NN  ->  (  seq 1 (  x.  , 
( NN  X.  {
1 } ) ) `
 ( # `  A
) )  =  1 )
4039adantr 463 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  (  seq 1 (  x.  , 
( NN  X.  {
1 } ) ) `
 ( # `  A
) )  =  1 )
4137, 40eqtrd 2423 . . . . . . 7  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  prod_ k  e.  A  1  =  1 )
4241ex 432 . . . . . 6  |-  ( (
# `  A )  e.  NN  ->  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  prod_ k  e.  A  1  =  1 ) )
4342exlimdv 1732 . . . . 5  |-  ( (
# `  A )  e.  NN  ->  ( E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A  ->  prod_ k  e.  A  1  =  1 ) )
4443imp 427 . . . 4  |-  ( ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A )  ->  prod_ k  e.  A 
1  =  1 )
4525, 44jaoi 377 . . 3  |-  ( ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) )  ->  prod_ k  e.  A  1  =  1 )
4628, 45syl 16 . 2  |-  ( A  e.  Fin  ->  prod_ k  e.  A  1  =  1 )
4727, 46jaoi 377 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 366    /\ wa 367    = wceq 1399   E.wex 1620    e. wcel 1826    =/= wne 2577    C_ wss 3389   (/)c0 3711   ifcif 3857   ~Pcpw 3927   {csn 3944   class class class wbr 4367    X. cxp 4911   dom cdm 4913   -1-1-onto->wf1o 5495   ` cfv 5496  (class class class)co 6196   Fincfn 7435   0cc0 9403   1c1 9404    x. cmul 9408   NNcn 10452   ZZcz 10781   ZZ>=cuz 11001   ...cfz 11593    seqcseq 12010   #chash 12307    ~~> cli 13309   prod_cprod 13714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-fzo 11718  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-prod 13715
This theorem is referenced by:  etransclem35  32218
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