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Theorem prodmolem2 13894
Description: Lemma for prodmo 13895. (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 995 . . 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 2872 . 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 5849 . . . . . 6  |-  ( m  =  w  ->  ( ZZ>=
`  m )  =  ( ZZ>= `  w )
)
43sseq2d 3470 . . . . 5  |-  ( m  =  w  ->  ( A  C_  ( ZZ>= `  m
)  <->  A  C_  ( ZZ>= `  w ) ) )
5 seqeq1 12154 . . . . . 6  |-  ( m  =  w  ->  seq m (  x.  ,  F )  =  seq w (  x.  ,  F ) )
65breq1d 4405 . . . . 5  |-  ( m  =  w  ->  (  seq m (  x.  ,  F )  ~~>  x  <->  seq w
(  x.  ,  F
)  ~~>  x ) )
74, 6anbi12d 709 . . . 4  |-  ( m  =  w  ->  (
( A  C_  ( ZZ>=
`  m )  /\  seq m (  x.  ,  F )  ~~>  x )  <-> 
( A  C_  ( ZZ>=
`  w )  /\  seq w (  x.  ,  F )  ~~>  x ) ) )
87cbvrexv 3035 . . 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 2975 . . . . 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 765 . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . 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 11146 . . . . . . . . . . . . . . . . 17  |-  ( ZZ>= `  w )  C_  ZZ
13 zssre 10912 . . . . . . . . . . . . . . . . 17  |-  ZZ  C_  RR
1412, 13sstri 3451 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= `  w )  C_  RR
1511, 14syl6ss 3454 . . . . . . . . . . . . . . 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 9696 . . . . . . . . . . . . . . 15  |-  <  Or  RR
17 soss 4762 . . . . . . . . . . . . . . 15  |-  ( A 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  A ) )
1815, 16, 17mpisyl 21 . . . . . . . . . . . . . 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 12123 . . . . . . . . . . . . . . 15  |-  ( 1 ... m )  e. 
Fin
20 ovex 6306 . . . . . . . . . . . . . . . . . 18  |-  ( 1 ... m )  e. 
_V
2120f1oen 7574 . . . . . . . . . . . . . . . . 17  |-  ( f : ( 1 ... m ) -1-1-onto-> A  ->  ( 1 ... m )  ~~  A )
2221ad2antll 727 . . . . . . . . . . . . . . . 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 7604 . . . . . . . . . . . . . . 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 7772 . . . . . . . . . . . . . . 15  |-  ( ( ( 1 ... m
)  e.  Fin  /\  A  ~~  ( 1 ... m ) )  ->  A  e.  Fin )
2519, 23, 24sylancr 661 . . . . . . . . . . . . . 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 12560 . . . . . . . . . . . . . 14  |-  ( (  <  Or  A  /\  A  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
2718, 25, 26syl2anc 659 . . . . . . . . . . . . 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 752 . . . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . . . 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 2402 . . . . . . . . . . . . . . . 16  |-  ( j  e.  NN  |->  [_ (
g `  j )  /  k ]_ B
)  =  ( j  e.  NN  |->  [_ (
g `  j )  /  k ]_ B
)
34 simplrr 763 . . . . . . . . . . . . . . . 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 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
) ) )  ->  w  e.  ZZ )
36 simplll 760 . . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . 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 13893 . . . . . . . . . . . . . . 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 613 . . . . . . . . . . . . . 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 1745 . . . . . . . . . . . . 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 13524 . . . . . . . . . . . 12  |-  ( (  seq w (  x.  ,  F )  ~~>  x  /\  seq w (  x.  ,  F )  ~~>  (  seq 1 (  x.  ,  G ) `  m
) )  ->  x  =  (  seq 1
(  x.  ,  G
) `  m )
)
4510, 43, 44syl2anc 659 . . . . . . . . . . 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 2417 . . . . . . . . . . 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 613 . . . . . . . . 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 429 . . . . . . . 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 1745 . . . . . . 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 601 . . . . . 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 2903 . . . . 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 435 . . 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 473 . 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 472 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 367    /\ w3a 974    = wceq 1405   E.wex 1633    e. wcel 1842    =/= wne 2598   E.wrex 2755   [_csb 3373    C_ wss 3414   ifcif 3885   class class class wbr 4395    |-> cmpt 4453    Or wor 4743   -1-1-onto->wf1o 5568   ` cfv 5569    Isom wiso 5570  (class class class)co 6278    ~~ cen 7551   Fincfn 7554   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523    x. cmul 9527    < clt 9658   NNcn 10576   ZZcz 10905   ZZ>=cuz 11127   ...cfz 11726    seqcseq 12151   #chash 12452    ~~> cli 13456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460
This theorem is referenced by:  prodmo  13895
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