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Theorem zfz1iso 10552
Description: A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)
Assertion
Ref Expression
zfz1iso  |-  ( ( A  C_  ZZ  /\  A  e.  Fin )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  A ) ) ,  A ) )
Distinct variable group:    A, f

Proof of Theorem zfz1iso
Dummy variables  n  x  a  k  m  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6623 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32adantl 275 . 2  |-  ( ( A  C_  ZZ  /\  A  e.  Fin )  ->  E. n  e.  om  A  ~~  n
)
4 simprlr 512 . . . 4  |-  ( ( n  e.  om  /\  ( ( A  C_  ZZ  /\  A  e.  Fin )  /\  A  ~~  n
) )  ->  A  e.  Fin )
5 breq2 3903 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( x 
~~  w  <->  x  ~~  (/) ) )
65anbi2d 459 . . . . . . . 8  |-  ( w  =  (/)  ->  ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  w )  <-> 
( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  (/) ) ) )
76imbi1d 230 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  w
)  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )  <->  ( (
( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  (/) )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) ) )
87albidv 1780 . . . . . 6  |-  ( w  =  (/)  ->  ( A. x ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  w )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )  <->  A. x
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  (/) )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) ) ) )
9 breq2 3903 . . . . . . . . 9  |-  ( w  =  k  ->  (
x  ~~  w  <->  x  ~~  k ) )
109anbi2d 459 . . . . . . . 8  |-  ( w  =  k  ->  (
( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  w
)  <->  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  k ) ) )
1110imbi1d 230 . . . . . . 7  |-  ( w  =  k  ->  (
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  w )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )  <->  ( (
( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) ) )
1211albidv 1780 . . . . . 6  |-  ( w  =  k  ->  ( A. x ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  w )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )  <->  A. x
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) ) ) )
13 breq2 3903 . . . . . . . . 9  |-  ( w  =  suc  k  -> 
( x  ~~  w  <->  x 
~~  suc  k )
)
1413anbi2d 459 . . . . . . . 8  |-  ( w  =  suc  k  -> 
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  w )  <->  ( (
x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) ) )
1514imbi1d 230 . . . . . . 7  |-  ( w  =  suc  k  -> 
( ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  w )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )  <->  ( (
( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k
)  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) ) ) )
1615albidv 1780 . . . . . 6  |-  ( w  =  suc  k  -> 
( A. x ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  w
)  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )  <->  A. x
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) ) )
17 breq2 3903 . . . . . . . . 9  |-  ( w  =  n  ->  (
x  ~~  w  <->  x  ~~  n ) )
1817anbi2d 459 . . . . . . . 8  |-  ( w  =  n  ->  (
( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  w
)  <->  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  n ) ) )
1918imbi1d 230 . . . . . . 7  |-  ( w  =  n  ->  (
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  w )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )  <->  ( (
( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  n )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) ) )
2019albidv 1780 . . . . . 6  |-  ( w  =  n  ->  ( A. x ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  w )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )  <->  A. x
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  n )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) ) ) )
21 iso0 5686 . . . . . . . . . 10  |-  (/)  Isom  <  ,  <  ( (/) ,  (/) )
22 en0 6657 . . . . . . . . . . . . . . . . 17  |-  ( x 
~~  (/)  <->  x  =  (/) )
2322biimpi 119 . . . . . . . . . . . . . . . 16  |-  ( x 
~~  (/)  ->  x  =  (/) )
2423fveq2d 5393 . . . . . . . . . . . . . . 15  |-  ( x 
~~  (/)  ->  ( `  x
)  =  ( `  (/) ) )
25 hash0 10511 . . . . . . . . . . . . . . 15  |-  ( `  (/) )  =  0
2624, 25syl6eq 2166 . . . . . . . . . . . . . 14  |-  ( x 
~~  (/)  ->  ( `  x
)  =  0 )
2726oveq2d 5758 . . . . . . . . . . . . 13  |-  ( x 
~~  (/)  ->  ( 1 ... ( `  x
) )  =  ( 1 ... 0 ) )
28 fz10 9794 . . . . . . . . . . . . 13  |-  ( 1 ... 0 )  =  (/)
2927, 28syl6eq 2166 . . . . . . . . . . . 12  |-  ( x 
~~  (/)  ->  ( 1 ... ( `  x
) )  =  (/) )
30 isoeq4 5673 . . . . . . . . . . . 12  |-  ( ( 1 ... ( `  x
) )  =  (/)  ->  ( (/)  Isom  <  ,  <  ( ( 1 ... ( `  x )
) ,  x )  <->  (/) 
Isom  <  ,  <  ( (/)
,  x ) ) )
3129, 30syl 14 . . . . . . . . . . 11  |-  ( x 
~~  (/)  ->  ( (/)  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
)  <->  (/)  Isom  <  ,  <  (
(/) ,  x )
) )
32 isoeq5 5674 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( (/)  Isom 
<  ,  <  ( (/) ,  x )  <->  (/)  Isom  <  ,  <  ( (/) ,  (/) ) ) )
3323, 32syl 14 . . . . . . . . . . 11  |-  ( x 
~~  (/)  ->  ( (/)  Isom  <  ,  <  ( (/) ,  x
)  <->  (/)  Isom  <  ,  <  (
(/) ,  (/) ) ) )
3431, 33bitrd 187 . . . . . . . . . 10  |-  ( x 
~~  (/)  ->  ( (/)  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
)  <->  (/)  Isom  <  ,  <  (
(/) ,  (/) ) ) )
3521, 34mpbiri 167 . . . . . . . . 9  |-  ( x 
~~  (/)  ->  (/)  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) )
36 0ex 4025 . . . . . . . . . 10  |-  (/)  e.  _V
37 isoeq1 5670 . . . . . . . . . 10  |-  ( f  =  (/)  ->  ( f 
Isom  <  ,  <  (
( 1 ... ( `  x ) ) ,  x )  <->  (/)  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) )
3836, 37spcev 2754 . . . . . . . . 9  |-  ( (/)  Isom 
<  ,  <  ( ( 1 ... ( `  x
) ) ,  x
)  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )
3935, 38syl 14 . . . . . . . 8  |-  ( x 
~~  (/)  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )
4039adantl 275 . . . . . . 7  |-  ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  (/) )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) )
4140ax-gen 1410 . . . . . 6  |-  A. x
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  (/) )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )
42 sseq1 3090 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
a  C_  ZZ  <->  x  C_  ZZ ) )
43 eleq1 2180 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
a  e.  Fin  <->  x  e.  Fin ) )
4442, 43anbi12d 464 . . . . . . . . . 10  |-  ( a  =  x  ->  (
( a  C_  ZZ  /\  a  e.  Fin )  <->  ( x  C_  ZZ  /\  x  e.  Fin ) ) )
45 breq1 3902 . . . . . . . . . 10  |-  ( a  =  x  ->  (
a  ~~  k  <->  x  ~~  k ) )
4644, 45anbi12d 464 . . . . . . . . 9  |-  ( a  =  x  ->  (
( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k
)  <->  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  k ) ) )
47 fveq2 5389 . . . . . . . . . . . . 13  |-  ( a  =  x  ->  ( `  a )  =  ( `  x ) )
4847oveq2d 5758 . . . . . . . . . . . 12  |-  ( a  =  x  ->  (
1 ... ( `  a
) )  =  ( 1 ... ( `  x
) ) )
49 isoeq4 5673 . . . . . . . . . . . 12  |-  ( ( 1 ... ( `  a
) )  =  ( 1 ... ( `  x
) )  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a )  <->  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  a ) ) )
5048, 49syl 14 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a )  <->  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  a ) ) )
51 isoeq5 5674 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  a )  <->  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) )
5250, 51bitrd 187 . . . . . . . . . 10  |-  ( a  =  x  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a )  <->  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) )
5352exbidv 1781 . . . . . . . . 9  |-  ( a  =  x  ->  ( E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  a
) ) ,  a )  <->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) )
5446, 53imbi12d 233 . . . . . . . 8  |-  ( a  =  x  ->  (
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) )  <->  ( (
( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) ) )
5554cbvalv 1871 . . . . . . 7  |-  ( A. a ( ( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) )  <->  A. x
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) ) )
56 simprll 511 . . . . . . . . . . . . 13  |-  ( ( ( k  e.  om  /\ 
A. a ( ( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  a
) ) ,  a ) ) )  /\  ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  ->  x  C_  ZZ )
57 zssq 9387 . . . . . . . . . . . . 13  |-  ZZ  C_  QQ
5856, 57sstrdi 3079 . . . . . . . . . . . 12  |-  ( ( ( k  e.  om  /\ 
A. a ( ( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  a
) ) ,  a ) ) )  /\  ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  ->  x  C_  QQ )
59 simprlr 512 . . . . . . . . . . . 12  |-  ( ( ( k  e.  om  /\ 
A. a ( ( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  a
) ) ,  a ) ) )  /\  ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  ->  x  e.  Fin )
60 vex 2663 . . . . . . . . . . . . . . . 16  |-  k  e. 
_V
61 nsuceq0g 4310 . . . . . . . . . . . . . . . 16  |-  ( k  e.  _V  ->  suc  k  =/=  (/) )
6260, 61ax-mp 5 . . . . . . . . . . . . . . 15  |-  suc  k  =/=  (/)
6362neii 2287 . . . . . . . . . . . . . 14  |-  -.  suc  k  =  (/)
64 simplrr 510 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  x  =  (/) )  ->  x  ~~  suc  k )
6564ensymd 6645 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  x  =  (/) )  ->  suc  k  ~~  x )
66 simpr 109 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  x  =  (/) )  ->  x  =  (/) )
6765, 66breqtrd 3924 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  x  =  (/) )  ->  suc  k  ~~  (/) )
68 en0 6657 . . . . . . . . . . . . . . . 16  |-  ( suc  k  ~~  (/)  <->  suc  k  =  (/) )
6967, 68sylib 121 . . . . . . . . . . . . . . 15  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  x  =  (/) )  ->  suc  k  =  (/) )
7069ex 114 . . . . . . . . . . . . . 14  |-  ( ( ( k  e.  om  /\ 
A. a ( ( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  a
) ) ,  a ) ) )  /\  ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  -> 
( x  =  (/)  ->  suc  k  =  (/) ) )
7163, 70mtoi 638 . . . . . . . . . . . . 13  |-  ( ( ( k  e.  om  /\ 
A. a ( ( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  a
) ) ,  a ) ) )  /\  ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  ->  -.  x  =  (/) )
7271neqned 2292 . . . . . . . . . . . 12  |-  ( ( ( k  e.  om  /\ 
A. a ( ( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  a
) ) ,  a ) ) )  /\  ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  ->  x  =/=  (/) )
73 fimaxq 10541 . . . . . . . . . . . 12  |-  ( ( x  C_  QQ  /\  x  e.  Fin  /\  x  =/=  (/) )  ->  E. m  e.  x  A. z  e.  x  z  <_  m )
7458, 59, 72, 73syl3anc 1201 . . . . . . . . . . 11  |-  ( ( ( k  e.  om  /\ 
A. a ( ( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  a
) ) ,  a ) ) )  /\  ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  ->  E. m  e.  x  A. z  e.  x  z  <_  m )
75 simplll 507 . . . . . . . . . . . 12  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  ( m  e.  x  /\  A. z  e.  x  z  <_  m ) )  ->  k  e.  om )
76 simpllr 508 . . . . . . . . . . . 12  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  ( m  e.  x  /\  A. z  e.  x  z  <_  m ) )  ->  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )
7756adantr 274 . . . . . . . . . . . 12  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  ( m  e.  x  /\  A. z  e.  x  z  <_  m ) )  ->  x  C_  ZZ )
7859adantr 274 . . . . . . . . . . . 12  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  ( m  e.  x  /\  A. z  e.  x  z  <_  m ) )  ->  x  e.  Fin )
79 simplrr 510 . . . . . . . . . . . 12  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  ( m  e.  x  /\  A. z  e.  x  z  <_  m ) )  ->  x  ~~  suc  k )
80 simprl 505 . . . . . . . . . . . 12  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  ( m  e.  x  /\  A. z  e.  x  z  <_  m ) )  ->  m  e.  x )
81 simprr 506 . . . . . . . . . . . 12  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  ( m  e.  x  /\  A. z  e.  x  z  <_  m ) )  ->  A. z  e.  x  z  <_  m )
8275, 76, 77, 78, 79, 80, 81zfz1isolem1 10551 . . . . . . . . . . 11  |-  ( ( ( ( k  e. 
om  /\  A. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  /\  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  /\  ( m  e.  x  /\  A. z  e.  x  z  <_  m ) )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )
8374, 82rexlimddv 2531 . . . . . . . . . 10  |-  ( ( ( k  e.  om  /\ 
A. a ( ( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  a
) ) ,  a ) ) )  /\  ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k ) )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) )
8483ex 114 . . . . . . . . 9  |-  ( ( k  e.  om  /\  A. a ( ( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  ->  ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) )
8584alrimiv 1830 . . . . . . . 8  |-  ( ( k  e.  om  /\  A. a ( ( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )  ->  A. x ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k
)  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) ) )
8685ex 114 . . . . . . 7  |-  ( k  e.  om  ->  ( A. a ( ( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) )  ->  A. x ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) ) )
8755, 86syl5bir 152 . . . . . 6  |-  ( k  e.  om  ->  ( A. x ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )  ->  A. x ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  suc  k )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) ) )
888, 12, 16, 20, 41, 87finds 4484 . . . . 5  |-  ( n  e.  om  ->  A. x
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  n )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) ) )
8988adantr 274 . . . 4  |-  ( ( n  e.  om  /\  ( ( A  C_  ZZ  /\  A  e.  Fin )  /\  A  ~~  n
) )  ->  A. x
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  n )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) ) )
90 simpr 109 . . . 4  |-  ( ( n  e.  om  /\  ( ( A  C_  ZZ  /\  A  e.  Fin )  /\  A  ~~  n
) )  ->  (
( A  C_  ZZ  /\  A  e.  Fin )  /\  A  ~~  n ) )
91 sseq1 3090 . . . . . . . 8  |-  ( x  =  A  ->  (
x  C_  ZZ  <->  A  C_  ZZ ) )
92 eleq1 2180 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  Fin  <->  A  e.  Fin ) )
9391, 92anbi12d 464 . . . . . . 7  |-  ( x  =  A  ->  (
( x  C_  ZZ  /\  x  e.  Fin )  <->  ( A  C_  ZZ  /\  A  e.  Fin ) ) )
94 breq1 3902 . . . . . . 7  |-  ( x  =  A  ->  (
x  ~~  n  <->  A  ~~  n ) )
9593, 94anbi12d 464 . . . . . 6  |-  ( x  =  A  ->  (
( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  n
)  <->  ( ( A 
C_  ZZ  /\  A  e. 
Fin )  /\  A  ~~  n ) ) )
96 fveq2 5389 . . . . . . . . . 10  |-  ( x  =  A  ->  ( `  x )  =  ( `  A ) )
9796oveq2d 5758 . . . . . . . . 9  |-  ( x  =  A  ->  (
1 ... ( `  x
) )  =  ( 1 ... ( `  A
) ) )
98 isoeq4 5673 . . . . . . . . 9  |-  ( ( 1 ... ( `  x
) )  =  ( 1 ... ( `  A
) )  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x )  <->  f  Isom  <  ,  <  ( ( 1 ... ( `  A
) ) ,  x
) ) )
9997, 98syl 14 . . . . . . . 8  |-  ( x  =  A  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x )  <->  f  Isom  <  ,  <  ( ( 1 ... ( `  A
) ) ,  x
) ) )
100 isoeq5 5674 . . . . . . . 8  |-  ( x  =  A  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( `  A ) ) ,  x )  <->  f  Isom  <  ,  <  ( ( 1 ... ( `  A
) ) ,  A
) ) )
10199, 100bitrd 187 . . . . . . 7  |-  ( x  =  A  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x )  <->  f  Isom  <  ,  <  ( ( 1 ... ( `  A
) ) ,  A
) ) )
102101exbidv 1781 . . . . . 6  |-  ( x  =  A  ->  ( E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
)  <->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  A
) ) ,  A
) ) )
10395, 102imbi12d 233 . . . . 5  |-  ( x  =  A  ->  (
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  n )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )  <->  ( (
( A  C_  ZZ  /\  A  e.  Fin )  /\  A  ~~  n )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  A
) ) ,  A
) ) ) )
104103spcgv 2747 . . . 4  |-  ( A  e.  Fin  ->  ( A. x ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  n )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )  -> 
( ( ( A 
C_  ZZ  /\  A  e. 
Fin )  /\  A  ~~  n )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  A ) ) ,  A ) ) ) )
1054, 89, 90, 104syl3c 63 . . 3  |-  ( ( n  e.  om  /\  ( ( A  C_  ZZ  /\  A  e.  Fin )  /\  A  ~~  n
) )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  A ) ) ,  A ) )
106105an12s 539 . 2  |-  ( ( ( A  C_  ZZ  /\  A  e.  Fin )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  A ) ) ,  A ) )
1073, 106rexlimddv 2531 1  |-  ( ( A  C_  ZZ  /\  A  e.  Fin )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  A ) ) ,  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1314    = wceq 1316   E.wex 1453    e. wcel 1465    =/= wne 2285   A.wral 2393   E.wrex 2394   _Vcvv 2660    C_ wss 3041   (/)c0 3333   class class class wbr 3899   suc csuc 4257   omcom 4474   ` cfv 5093    Isom wiso 5094  (class class class)co 5742    ~~ cen 6600   Fincfn 6602   0cc0 7588   1c1 7589    < clt 7768    <_ cle 7769   ZZcz 9022   QQcq 9379   ...cfz 9758  ♯chash 10489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-isom 5102  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-frec 6256  df-1o 6281  df-oadd 6285  df-er 6397  df-en 6603  df-dom 6604  df-fin 6605  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-ihash 10490
This theorem is referenced by:  summodclem2  11119  zsumdc  11121
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