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Theorem prodmolem2 27451
Description: Lemma for prodmo 27452. (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 986 . . 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 2826 . 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 5694 . . . . . 6  |-  ( m  =  w  ->  ( ZZ>=
`  m )  =  ( ZZ>= `  w )
)
43sseq2d 3387 . . . . 5  |-  ( m  =  w  ->  ( A  C_  ( ZZ>= `  m
)  <->  A  C_  ( ZZ>= `  w ) ) )
5 seqeq1 11812 . . . . . 6  |-  ( m  =  w  ->  seq m (  x.  ,  F )  =  seq w (  x.  ,  F ) )
65breq1d 4305 . . . . 5  |-  ( m  =  w  ->  (  seq m (  x.  ,  F )  ~~>  x  <->  seq w
(  x.  ,  F
)  ~~>  x ) )
74, 6anbi12d 710 . . . 4  |-  ( m  =  w  ->  (
( A  C_  ( ZZ>=
`  m )  /\  seq m (  x.  ,  F )  ~~>  x )  <-> 
( A  C_  ( ZZ>=
`  w )  /\  seq w (  x.  ,  F )  ~~>  x ) ) )
87cbvrexv 2951 . . 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 2891 . . . . 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 762 . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . 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 10883 . . . . . . . . . . . . . . . . 17  |-  ( ZZ>= `  w )  C_  ZZ
13 zssre 10656 . . . . . . . . . . . . . . . . 17  |-  ZZ  C_  RR
1412, 13sstri 3368 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= `  w )  C_  RR
1511, 14syl6ss 3371 . . . . . . . . . . . . . . 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 9458 . . . . . . . . . . . . . . 15  |-  <  Or  RR
17 soss 4662 . . . . . . . . . . . . . . 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 11797 . . . . . . . . . . . . . . 15  |-  ( 1 ... m )  e. 
Fin
20 ovex 6119 . . . . . . . . . . . . . . . . . 18  |-  ( 1 ... m )  e. 
_V
2120f1oen 7333 . . . . . . . . . . . . . . . . 17  |-  ( f : ( 1 ... m ) -1-1-onto-> A  ->  ( 1 ... m )  ~~  A )
2221ad2antll 728 . . . . . . . . . . . . . . . 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 7363 . . . . . . . . . . . . . . 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 7533 . . . . . . . . . . . . . . 15  |-  ( ( ( 1 ... m
)  e.  Fin  /\  A  ~~  ( 1 ... m ) )  ->  A  e.  Fin )
2519, 23, 24sylancr 663 . . . . . . . . . . . . . 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 12218 . . . . . . . . . . . . . 14  |-  ( (  <  Or  A  /\  A  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
2718, 25, 26syl2anc 661 . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . . 16  |-  ( j  e.  NN  |->  [_ (
g `  j )  /  k ]_ B
)  =  ( j  e.  NN  |->  [_ (
g `  j )  /  k ]_ B
)
34 simplrr 760 . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . 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 27450 . . . . . . . . . . . . . . 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 615 . . . . . . . . . . . . . 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 1690 . . . . . . . . . . . . 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 13033 . . . . . . . . . . . 12  |-  ( (  seq w (  x.  ,  F )  ~~>  x  /\  seq w (  x.  ,  F )  ~~>  (  seq 1 (  x.  ,  G ) `  m
) )  ->  x  =  (  seq 1
(  x.  ,  G
) `  m )
)
4510, 43, 44syl2anc 661 . . . . . . . . . . 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 2452 . . . . . . . . . . 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 615 . . . . . . . . 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 ) ) )
4948impd 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 1690 . . . . . . 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 603 . . . . . 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 2851 . . . . 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 475 . 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 474 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 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2609   E.wrex 2719   [_csb 3291    C_ wss 3331   ifcif 3794   class class class wbr 4295    e. cmpt 4353    Or wor 4643   -1-1-onto->wf1o 5420   ` cfv 5421    Isom wiso 5422  (class class class)co 6094    ~~ cen 7310   Fincfn 7313   CCcc 9283   RRcr 9284   0cc0 9285   1c1 9286    x. cmul 9290    < clt 9421   NNcn 10325   ZZcz 10649   ZZ>=cuz 10864   ...cfz 11440    seqcseq 11809   #chash 12106    ~~> cli 12965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-oi 7727  df-card 8112  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-rp 10995  df-fz 11441  df-fzo 11552  df-seq 11810  df-exp 11869  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969
This theorem is referenced by:  prodmo  27452
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