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Theorem zfz1iso 10535
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 6621 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32adantl 273 . 2  |-  ( ( A  C_  ZZ  /\  A  e.  Fin )  ->  E. n  e.  om  A  ~~  n
)
4 simprlr 510 . . . 4  |-  ( ( n  e.  om  /\  ( ( A  C_  ZZ  /\  A  e.  Fin )  /\  A  ~~  n
) )  ->  A  e.  Fin )
5 breq2 3901 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( x 
~~  w  <->  x  ~~  (/) ) )
65anbi2d 457 . . . . . . . 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 1778 . . . . . 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 3901 . . . . . . . . 9  |-  ( w  =  k  ->  (
x  ~~  w  <->  x  ~~  k ) )
109anbi2d 457 . . . . . . . 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 1778 . . . . . 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 3901 . . . . . . . . 9  |-  ( w  =  suc  k  -> 
( x  ~~  w  <->  x 
~~  suc  k )
)
1413anbi2d 457 . . . . . . . 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 1778 . . . . . 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 3901 . . . . . . . . 9  |-  ( w  =  n  ->  (
x  ~~  w  <->  x  ~~  n ) )
1817anbi2d 457 . . . . . . . 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 1778 . . . . . 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 5684 . . . . . . . . . 10  |-  (/)  Isom  <  ,  <  ( (/) ,  (/) )
22 en0 6655 . . . . . . . . . . . . . . . . 17  |-  ( x 
~~  (/)  <->  x  =  (/) )
2322biimpi 119 . . . . . . . . . . . . . . . 16  |-  ( x 
~~  (/)  ->  x  =  (/) )
2423fveq2d 5391 . . . . . . . . . . . . . . 15  |-  ( x 
~~  (/)  ->  ( `  x
)  =  ( `  (/) ) )
25 hash0 10494 . . . . . . . . . . . . . . 15  |-  ( `  (/) )  =  0
2624, 25syl6eq 2164 . . . . . . . . . . . . . 14  |-  ( x 
~~  (/)  ->  ( `  x
)  =  0 )
2726oveq2d 5756 . . . . . . . . . . . . 13  |-  ( x 
~~  (/)  ->  ( 1 ... ( `  x
) )  =  ( 1 ... 0 ) )
28 fz10 9777 . . . . . . . . . . . . 13  |-  ( 1 ... 0 )  =  (/)
2927, 28syl6eq 2164 . . . . . . . . . . . 12  |-  ( x 
~~  (/)  ->  ( 1 ... ( `  x
) )  =  (/) )
30 isoeq4 5671 . . . . . . . . . . . 12  |-  ( ( 1 ... ( `  x
) )  =  (/)  ->  ( (/)  Isom  <  ,  <  ( ( 1 ... ( `  x )
) ,  x )  <->  (/) 
Isom  <  ,  <  ( (/)
,  x ) ) )
3129, 30syl 14 . . . . . . . . . . 11  |-  ( x 
~~  (/)  ->  ( (/)  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
)  <->  (/)  Isom  <  ,  <  (
(/) ,  x )
) )
32 isoeq5 5672 . . . . . . . . . . . 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 4023 . . . . . . . . . 10  |-  (/)  e.  _V
37 isoeq1 5668 . . . . . . . . . 10  |-  ( f  =  (/)  ->  ( f 
Isom  <  ,  <  (
( 1 ... ( `  x ) ) ,  x )  <->  (/)  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) ) )
3836, 37spcev 2752 . . . . . . . . 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 273 . . . . . . 7  |-  ( ( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  (/) )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  x
) ) ,  x
) )
4140ax-gen 1408 . . . . . 6  |-  A. x
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  (/) )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) )
42 sseq1 3088 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
a  C_  ZZ  <->  x  C_  ZZ ) )
43 eleq1 2178 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
a  e.  Fin  <->  x  e.  Fin ) )
4442, 43anbi12d 462 . . . . . . . . . 10  |-  ( a  =  x  ->  (
( a  C_  ZZ  /\  a  e.  Fin )  <->  ( x  C_  ZZ  /\  x  e.  Fin ) ) )
45 breq1 3900 . . . . . . . . . 10  |-  ( a  =  x  ->  (
a  ~~  k  <->  x  ~~  k ) )
4644, 45anbi12d 462 . . . . . . . . 9  |-  ( a  =  x  ->  (
( ( a  C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k
)  <->  ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  k ) ) )
47 fveq2 5387 . . . . . . . . . . . . 13  |-  ( a  =  x  ->  ( `  a )  =  ( `  x ) )
4847oveq2d 5756 . . . . . . . . . . . 12  |-  ( a  =  x  ->  (
1 ... ( `  a
) )  =  ( 1 ... ( `  x
) ) )
49 isoeq4 5671 . . . . . . . . . . . 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 5672 . . . . . . . . . . 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 1779 . . . . . . . . 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 1869 . . . . . . 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 509 . . . . . . . . . . . . 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 9371 . . . . . . . . . . . . 13  |-  ZZ  C_  QQ
5856, 57syl6ss 3077 . . . . . . . . . . . 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 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 ) )  ->  x  e.  Fin )
60 vex 2661 . . . . . . . . . . . . . . . 16  |-  k  e. 
_V
61 nsuceq0g 4308 . . . . . . . . . . . . . . . 16  |-  ( k  e.  _V  ->  suc  k  =/=  (/) )
6260, 61ax-mp 5 . . . . . . . . . . . . . . 15  |-  suc  k  =/=  (/)
6362neii 2285 . . . . . . . . . . . . . 14  |-  -.  suc  k  =  (/)
64 simplrr 508 . . . . . . . . . . . . . . . . . 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 6643 . . . . . . . . . . . . . . . . 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 3922 . . . . . . . . . . . . . . . 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 6655 . . . . . . . . . . . . . . . 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 636 . . . . . . . . . . . . 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 2290 . . . . . . . . . . . 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 10524 . . . . . . . . . . . 12  |-  ( ( x  C_  QQ  /\  x  e.  Fin  /\  x  =/=  (/) )  ->  E. m  e.  x  A. z  e.  x  z  <_  m )
7458, 59, 72, 73syl3anc 1199 . . . . . . . . . . 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 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 ) )  ->  k  e.  om )
76 simpllr 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. a
( ( ( a 
C_  ZZ  /\  a  e.  Fin )  /\  a  ~~  k )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  a ) ) ,  a ) ) )
7756adantr 272 . . . . . . . . . . . 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 272 . . . . . . . . . . . 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 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 ) )  ->  x  ~~  suc  k )
80 simprl 503 . . . . . . . . . . . 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 504 . . . . . . . . . . . 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 10534 . . . . . . . . . . 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 2529 . . . . . . . . . 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 1828 . . . . . . . 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 4482 . . . . 5  |-  ( n  e.  om  ->  A. x
( ( ( x 
C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  n )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  x ) ) ,  x ) ) )
8988adantr 272 . . . 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 3088 . . . . . . . 8  |-  ( x  =  A  ->  (
x  C_  ZZ  <->  A  C_  ZZ ) )
92 eleq1 2178 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  Fin  <->  A  e.  Fin ) )
9391, 92anbi12d 462 . . . . . . 7  |-  ( x  =  A  ->  (
( x  C_  ZZ  /\  x  e.  Fin )  <->  ( A  C_  ZZ  /\  A  e.  Fin ) ) )
94 breq1 3900 . . . . . . 7  |-  ( x  =  A  ->  (
x  ~~  n  <->  A  ~~  n ) )
9593, 94anbi12d 462 . . . . . 6  |-  ( x  =  A  ->  (
( ( x  C_  ZZ  /\  x  e.  Fin )  /\  x  ~~  n
)  <->  ( ( A 
C_  ZZ  /\  A  e. 
Fin )  /\  A  ~~  n ) ) )
96 fveq2 5387 . . . . . . . . . 10  |-  ( x  =  A  ->  ( `  x )  =  ( `  A ) )
9796oveq2d 5756 . . . . . . . . 9  |-  ( x  =  A  ->  (
1 ... ( `  x
) )  =  ( 1 ... ( `  A
) ) )
98 isoeq4 5671 . . . . . . . . 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 5672 . . . . . . . 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 1779 . . . . . 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 2745 . . . 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 537 . 2  |-  ( ( ( A  C_  ZZ  /\  A  e.  Fin )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( `  A ) ) ,  A ) )
1073, 106rexlimddv 2529 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 1312    = wceq 1314   E.wex 1451    e. wcel 1463    =/= wne 2283   A.wral 2391   E.wrex 2392   _Vcvv 2658    C_ wss 3039   (/)c0 3331   class class class wbr 3897   suc csuc 4255   omcom 4472   ` cfv 5091    Isom wiso 5092  (class class class)co 5740    ~~ cen 6598   Fincfn 6600   0cc0 7584   1c1 7585    < clt 7764    <_ cle 7765   ZZcz 9008   QQcq 9363   ...cfz 9741  ♯chash 10472
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-fz 9742  df-ihash 10473
This theorem is referenced by:  summodclem2  11102  zsumdc  11104
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