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Theorem prodmolem2 27295
Description: Lemma for prodmo 27296. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
prodmo.1  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
prodmo.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
prodmo.3  |-  G  =  ( j  e.  NN  |->  [_ ( f `  j
)  /  k ]_ B )
Assertion
Ref Expression
prodmolem2  |-  ( (
ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  E. n  e.  (
ZZ>= `  m ) E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  seq m (  x.  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq 1 (  x.  ,  G ) `  m ) )  ->  x  =  z )
)
Distinct variable groups:    A, k, n    k, F, n    ph, k, n    A, f, j, m    B, j    f, F, j, k, m    ph, f    x, f    z, f    j, G    j, k, m, ph    x, j    k, m, x    ph, m    x, m    z, m
Allowed substitution hints:    ph( x, y, z)    A( x, y, z)    B( x, y, z, f, k, m, n)    F( x, y, z)    G( x, y, z, f, k, m, n)

Proof of Theorem prodmolem2
Dummy variables  g  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpb 979 . . 3  |-  ( ( A  C_  ( ZZ>= `  m )  /\  E. n  e.  ( ZZ>= `  m ) E. y
( y  =/=  0  /\  seq n (  x.  ,  F )  ~~>  y )  /\  seq m (  x.  ,  F )  ~~>  x )  ->  ( A  C_  ( ZZ>= `  m
)  /\  seq m
(  x.  ,  F
)  ~~>  x ) )
21reximi 2813 . 2  |-  ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  E. n  e.  ( ZZ>= `  m ) E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  seq m (  x.  ,  F )  ~~>  x )  ->  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  x.  ,  F )  ~~>  x ) )
3 fveq2 5679 . . . . . 6  |-  ( m  =  w  ->  ( ZZ>=
`  m )  =  ( ZZ>= `  w )
)
43sseq2d 3372 . . . . 5  |-  ( m  =  w  ->  ( A  C_  ( ZZ>= `  m
)  <->  A  C_  ( ZZ>= `  w ) ) )
5 seqeq1 11793 . . . . . 6  |-  ( m  =  w  ->  seq m (  x.  ,  F )  =  seq w (  x.  ,  F ) )
65breq1d 4290 . . . . 5  |-  ( m  =  w  ->  (  seq m (  x.  ,  F )  ~~>  x  <->  seq w
(  x.  ,  F
)  ~~>  x ) )
74, 6anbi12d 703 . . . 4  |-  ( m  =  w  ->  (
( A  C_  ( ZZ>=
`  m )  /\  seq m (  x.  ,  F )  ~~>  x )  <-> 
( A  C_  ( ZZ>=
`  w )  /\  seq w (  x.  ,  F )  ~~>  x ) ) )
87cbvrexv 2938 . . 3  |-  ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq m
(  x.  ,  F
)  ~~>  x )  <->  E. w  e.  ZZ  ( A  C_  ( ZZ>= `  w )  /\  seq w (  x.  ,  F )  ~~>  x ) )
9 reeanv 2878 . . . . 5  |-  ( E. w  e.  ZZ  E. m  e.  NN  (
( A  C_  ( ZZ>=
`  w )  /\  seq w (  x.  ,  F )  ~~>  x )  /\  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq 1 (  x.  ,  G ) `  m
) ) )  <->  ( E. w  e.  ZZ  ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  z  =  (  seq 1 (  x.  ,  G ) `  m ) ) ) )
10 simprlr 755 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  seq w (  x.  ,  F )  ~~>  x )
11 simprll 754 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  A  C_  ( ZZ>= `  w
) )
12 uzssz 10868 . . . . . . . . . . . . . . . . 17  |-  ( ZZ>= `  w )  C_  ZZ
13 zssre 10641 . . . . . . . . . . . . . . . . 17  |-  ZZ  C_  RR
1412, 13sstri 3353 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= `  w )  C_  RR
1511, 14syl6ss 3356 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  A  C_  RR )
16 ltso 9443 . . . . . . . . . . . . . . 15  |-  <  Or  RR
17 soss 4646 . . . . . . . . . . . . . . 15  |-  ( A 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  A ) )
1815, 16, 17mpisyl 18 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  <  Or  A )
19 fzfi 11778 . . . . . . . . . . . . . . 15  |-  ( 1 ... m )  e. 
Fin
20 ovex 6105 . . . . . . . . . . . . . . . . . 18  |-  ( 1 ... m )  e. 
_V
2120f1oen 7318 . . . . . . . . . . . . . . . . 17  |-  ( f : ( 1 ... m ) -1-1-onto-> A  ->  ( 1 ... m )  ~~  A )
2221ad2antll 721 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  -> 
( 1 ... m
)  ~~  A )
2322ensymd 7348 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  A  ~~  ( 1 ... m ) )
24 enfii 7518 . . . . . . . . . . . . . . 15  |-  ( ( ( 1 ... m
)  e.  Fin  /\  A  ~~  ( 1 ... m ) )  ->  A  e.  Fin )
2519, 23, 24sylancr 656 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  A  e.  Fin )
26 fz1iso 12199 . . . . . . . . . . . . . 14  |-  ( (  <  Or  A  /\  A  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
2718, 25, 26syl2anc 654 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
28 prodmo.1 . . . . . . . . . . . . . . . 16  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
29 simpll 746 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  ->  ph )
30 prodmo.2 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
3129, 30sylan 468 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( w  e.  ZZ  /\  m  e.  NN ) )  /\  ( ( ( A  C_  ( ZZ>=
`  w )  /\  seq w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  /\  k  e.  A )  ->  B  e.  CC )
32 prodmo.3 . . . . . . . . . . . . . . . 16  |-  G  =  ( j  e.  NN  |->  [_ ( f `  j
)  /  k ]_ B )
33 eqid 2433 . . . . . . . . . . . . . . . 16  |-  ( j  e.  NN  |->  [_ (
g `  j )  /  k ]_ B
)  =  ( j  e.  NN  |->  [_ (
g `  j )  /  k ]_ B
)
34 simplrr 753 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  ->  m  e.  NN )
35 simplrl 752 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  ->  w  e.  ZZ )
36 simplll 750 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  C_  ( ZZ>= `  w )  /\  seq w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )  ->  A  C_  ( ZZ>= `  w )
)
3736adantl 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  ->  A  C_  ( ZZ>= `  w
) )
38 simprlr 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  -> 
f : ( 1 ... m ) -1-1-onto-> A )
39 simprr 749 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  -> 
g  Isom  <  ,  <  ( ( 1 ... ( # `
 A ) ) ,  A ) )
4028, 31, 32, 33, 34, 35, 37, 38, 39prodmolem2a 27294 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( (
( A  C_  ( ZZ>=
`  w )  /\  seq w (  x.  ,  F )  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) ) )  ->  seq w (  x.  ,  F )  ~~>  (  seq 1 (  x.  ,  G ) `  m
) )
4140expr 610 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  -> 
( g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
)  ->  seq w
(  x.  ,  F
)  ~~>  (  seq 1
(  x.  ,  G
) `  m )
) )
4241exlimdv 1689 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  -> 
( E. g  g 
Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A )  ->  seq w (  x.  ,  F )  ~~>  (  seq 1 (  x.  ,  G ) `  m
) ) )
4327, 42mpd 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  seq w (  x.  ,  F )  ~~>  (  seq 1 (  x.  ,  G ) `  m
) )
44 climuni 13014 . . . . . . . . . . . 12  |-  ( (  seq w (  x.  ,  F )  ~~>  x  /\  seq w (  x.  ,  F )  ~~>  (  seq 1 (  x.  ,  G ) `  m
) )  ->  x  =  (  seq 1
(  x.  ,  G
) `  m )
)
4510, 43, 44syl2anc 654 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  x  =  (  seq 1 (  x.  ,  G ) `  m
) )
46 eqeq2 2442 . . . . . . . . . . 11  |-  ( z  =  (  seq 1
(  x.  ,  G
) `  m )  ->  ( x  =  z  <-> 
x  =  (  seq 1 (  x.  ,  G ) `  m
) ) )
4745, 46syl5ibrcom 222 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( ( A  C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  -> 
( z  =  (  seq 1 (  x.  ,  G ) `  m )  ->  x  =  z ) )
4847expr 610 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( A  C_  ( ZZ>= `  w )  /\  seq w (  x.  ,  F )  ~~>  x ) )  ->  ( f : ( 1 ... m ) -1-1-onto-> A  ->  ( z  =  (  seq 1
(  x.  ,  G
) `  m )  ->  x  =  z ) ) )
4948imp3a 431 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( A  C_  ( ZZ>= `  w )  /\  seq w (  x.  ,  F )  ~~>  x ) )  ->  ( (
f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq 1 (  x.  ,  G ) `  m
) )  ->  x  =  z ) )
5049exlimdv 1689 . . . . . . 7  |-  ( ( ( ph  /\  (
w  e.  ZZ  /\  m  e.  NN )
)  /\  ( A  C_  ( ZZ>= `  w )  /\  seq w (  x.  ,  F )  ~~>  x ) )  ->  ( E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  z  =  (  seq 1 (  x.  ,  G ) `  m ) )  ->  x  =  z )
)
5150expimpd 598 . . . . . 6  |-  ( (
ph  /\  ( w  e.  ZZ  /\  m  e.  NN ) )  -> 
( ( ( A 
C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  z  =  (  seq 1 (  x.  ,  G ) `  m ) ) )  ->  x  =  z ) )
5251rexlimdvva 2838 . . . . 5  |-  ( ph  ->  ( E. w  e.  ZZ  E. m  e.  NN  ( ( A 
C_  ( ZZ>= `  w
)  /\  seq w
(  x.  ,  F
)  ~~>  x )  /\  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  z  =  (  seq 1 (  x.  ,  G ) `  m ) ) )  ->  x  =  z ) )
539, 52syl5bir 218 . . . 4  |-  ( ph  ->  ( ( E. w  e.  ZZ  ( A  C_  ( ZZ>= `  w )  /\  seq w (  x.  ,  F )  ~~>  x )  /\  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq 1 (  x.  ,  G ) `  m
) ) )  ->  x  =  z )
)
5453expdimp 437 . . 3  |-  ( (
ph  /\  E. w  e.  ZZ  ( A  C_  ( ZZ>= `  w )  /\  seq w (  x.  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq 1 (  x.  ,  G ) `  m ) )  ->  x  =  z )
)
558, 54sylan2b 472 . 2  |-  ( (
ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  x.  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq 1 (  x.  ,  G ) `  m ) )  ->  x  =  z )
)
562, 55sylan2 471 1  |-  ( (
ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  E. n  e.  (
ZZ>= `  m ) E. y ( y  =/=  0  /\  seq n
(  x.  ,  F
)  ~~>  y )  /\  seq m (  x.  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq 1 (  x.  ,  G ) `  m ) )  ->  x  =  z )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 958    = wceq 1362   E.wex 1589    e. wcel 1755    =/= wne 2596   E.wrex 2706   [_csb 3276    C_ wss 3316   ifcif 3779   class class class wbr 4280    e. cmpt 4338    Or wor 4627   -1-1-onto->wf1o 5405   ` cfv 5406    Isom wiso 5407  (class class class)co 6080    ~~ cen 7295   Fincfn 7298   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    x. cmul 9275    < clt 9406   NNcn 10310   ZZcz 10634   ZZ>=cuz 10849   ...cfz 11424    seqcseq 11790   #chash 12087    ~~> cli 12946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950
This theorem is referenced by:  prodmo  27296
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