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Theorem prod1 25275
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 2438 . . . 4  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 simpr 449 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  M  e.  ZZ )
3 ax-1ne0 9064 . . . . 5  |-  1  =/=  0
43a1i 11 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  1  =/=  0 )
51prodfclim1 25226 . . . . 5  |-  ( M  e.  ZZ  ->  seq  M (  x.  ,  ( ( ZZ>= `  M )  X.  { 1 } ) )  ~~>  1 )
65adantl 454 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  seq  M (  x.  ,  ( ( ZZ>= `  M )  X.  { 1 } ) )  ~~>  1 )
7 simpl 445 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  A  C_  ( ZZ>= `  M )
)
8 1ex 9091 . . . . . . 7  |-  1  e.  _V
98fvconst2 5950 . . . . . 6  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 1 } ) `
 k )  =  1 )
10 ifid 3773 . . . . . 6  |-  if ( k  e.  A , 
1 ,  1 )  =  1
119, 10syl6eqr 2488 . . . . 5  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( (
( ZZ>= `  M )  X.  { 1 } ) `
 k )  =  if ( k  e.  A ,  1 ,  1 ) )
1211adantl 454 . . . 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 9053 . . . . 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 25270 . . 3  |-  ( ( A  C_  ( ZZ>= `  M )  /\  M  e.  ZZ )  ->  prod_ k  e.  A 1  =  1 )
16 uzf 10496 . . . . . . . . 9  |-  ZZ>= : ZZ --> ~P ZZ
1716fdmi 5599 . . . . . . . 8  |-  dom  ZZ>=  =  ZZ
1817eleq2i 2502 . . . . . . 7  |-  ( M  e.  dom  ZZ>=  <->  M  e.  ZZ )
19 ndmfv 5758 . . . . . . 7  |-  ( -.  M  e.  dom  ZZ>=  -> 
( ZZ>= `  M )  =  (/) )
2018, 19sylnbir 300 . . . . . 6  |-  ( -.  M  e.  ZZ  ->  (
ZZ>= `  M )  =  (/) )
2120sseq2d 3378 . . . . 5  |-  ( -.  M  e.  ZZ  ->  ( A  C_  ( ZZ>= `  M )  <->  A  C_  (/) ) )
2221biimpac 474 . . . 4  |-  ( ( A  C_  ( ZZ>= `  M )  /\  -.  M  e.  ZZ )  ->  A  C_  (/) )
23 ss0 3660 . . . 4  |-  ( A 
C_  (/)  ->  A  =  (/) )
24 prodeq1 25240 . . . . 5  |-  ( A  =  (/)  ->  prod_ k  e.  A 1  =  prod_ k  e.  (/) 1 )
25 prod0 25274 . . . . 5  |-  prod_ k  e.  (/) 1  =  1
2624, 25syl6eq 2486 . . . 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 768 . 2  |-  ( A 
C_  ( ZZ>= `  M
)  ->  prod_ k  e.  A 1  =  1 )
29 fz1f1o 12509 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
30 eqidd 2439 . . . . . . . . 9  |-  ( k  =  ( f `  j )  ->  1  =  1 )
31 simpl 445 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  ( # `
 A )  e.  NN )
32 simpr 449 . . . . . . . . 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 11085 . . . . . . . . . . 11  |-  ( j  e.  ( 1 ... ( # `  A
) )  ->  j  e.  NN )
358fvconst2 5950 . . . . . . . . . . 11  |-  ( j  e.  NN  ->  (
( NN  X.  {
1 } ) `  j )  =  1 )
3634, 35syl 16 . . . . . . . . . 10  |-  ( j  e.  ( 1 ... ( # `  A
) )  ->  (
( NN  X.  {
1 } ) `  j )  =  1 )
3736adantl 454 . . . . . . . . 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 25272 . . . . . . . 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 10526 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
4039prodf1 25224 . . . . . . . . 9  |-  ( (
# `  A )  e.  NN  ->  (  seq  1 (  x.  , 
( NN  X.  {
1 } ) ) `
 ( # `  A
) )  =  1 )
4140adantr 453 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  (  seq  1 (  x.  , 
( NN  X.  {
1 } ) ) `
 ( # `  A
) )  =  1 )
4238, 41eqtrd 2470 . . . . . . 7  |-  ( ( ( # `  A
)  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )  ->  prod_ k  e.  A 1  =  1 )
4342ex 425 . . . . . 6  |-  ( (
# `  A )  e.  NN  ->  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  prod_ k  e.  A 1  =  1 ) )
4443exlimdv 1647 . . . . 5  |-  ( (
# `  A )  e.  NN  ->  ( E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A  ->  prod_ k  e.  A 1  =  1 ) )
4544imp 420 . . . 4  |-  ( ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A )  ->  prod_ k  e.  A
1  =  1 )
4626, 45jaoi 370 . . 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 370 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 359    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601    C_ wss 3322   (/)c0 3630   ifcif 3741   ~Pcpw 3801   {csn 3816   class class class wbr 4215    X. cxp 4879   dom cdm 4881   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084   Fincfn 7112   CCcc 8993   0cc0 8995   1c1 8996    x. cmul 9000   NNcn 10005   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048    seq cseq 11328   #chash 11623    ~~> cli 12283   prod_cprod 25236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-prod 25237
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