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Theorem fzisoeu 37606
Description: A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 12666 for the base index and also states the uniqueness condition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fzisoeu.h  |-  ( ph  ->  H  e.  Fin )
fzisoeu.or  |-  ( ph  ->  <  Or  H )
fzisoeu.m  |-  ( ph  ->  M  e.  ZZ )
fzisoeu.4  |-  N  =  ( ( # `  H
)  +  ( M  -  1 ) )
Assertion
Ref Expression
fzisoeu  |-  ( ph  ->  E! f  f  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
Distinct variable groups:    f, H    f, M    f, N
Allowed substitution hint:    ph( f)

Proof of Theorem fzisoeu
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzssz 11827 . . . . . . . . 9  |-  ( M ... N )  C_  ZZ
2 zssre 10968 . . . . . . . . 9  |-  ZZ  C_  RR
31, 2sstri 3427 . . . . . . . 8  |-  ( M ... N )  C_  RR
4 ltso 9732 . . . . . . . 8  |-  <  Or  RR
5 soss 4778 . . . . . . . 8  |-  ( ( M ... N ) 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  ( M ... N ) ) )
63, 4, 5mp2 9 . . . . . . 7  |-  <  Or  ( M ... N )
7 fzfi 12223 . . . . . . 7  |-  ( M ... N )  e. 
Fin
8 fz1iso 12666 . . . . . . 7  |-  ( (  <  Or  ( M ... N )  /\  ( M ... N )  e.  Fin )  ->  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) ) )
96, 7, 8mp2an 686 . . . . . 6  |-  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `
 ( M ... N ) ) ) ,  ( M ... N ) )
10 fzisoeu.4 . . . . . . . . . . . . . . . 16  |-  N  =  ( ( # `  H
)  +  ( M  -  1 ) )
11 fveq2 5879 . . . . . . . . . . . . . . . . . 18  |-  ( H  =  (/)  ->  ( # `  H )  =  (
# `  (/) ) )
12 hash0 12586 . . . . . . . . . . . . . . . . . 18  |-  ( # `  (/) )  =  0
1311, 12syl6eq 2521 . . . . . . . . . . . . . . . . 17  |-  ( H  =  (/)  ->  ( # `  H )  =  0 )
1413oveq1d 6323 . . . . . . . . . . . . . . . 16  |-  ( H  =  (/)  ->  ( (
# `  H )  +  ( M  - 
1 ) )  =  ( 0  +  ( M  -  1 ) ) )
1510, 14syl5eq 2517 . . . . . . . . . . . . . . 15  |-  ( H  =  (/)  ->  N  =  ( 0  +  ( M  -  1 ) ) )
1615oveq2d 6324 . . . . . . . . . . . . . 14  |-  ( H  =  (/)  ->  ( M ... N )  =  ( M ... (
0  +  ( M  -  1 ) ) ) )
1716adantl 473 . . . . . . . . . . . . 13  |-  ( (
ph  /\  H  =  (/) )  ->  ( M ... N )  =  ( M ... ( 0  +  ( M  - 
1 ) ) ) )
18 fzisoeu.m . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  M  e.  ZZ )
1918zcnd 11064 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  M  e.  CC )
20 1cnd 9677 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  1  e.  CC )
2119, 20subcld 10005 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( M  -  1 )  e.  CC )
2221addid2d 9852 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 0  +  ( M  -  1 ) )  =  ( M  -  1 ) )
2322oveq2d 6324 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( M ... (
0  +  ( M  -  1 ) ) )  =  ( M ... ( M  - 
1 ) ) )
2418zred 11063 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  M  e.  RR )
2524ltm1d 10561 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( M  -  1 )  <  M )
26 peano2zm 11004 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
2718, 26syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
28 fzn 11841 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  ZZ  /\  ( M  -  1
)  e.  ZZ )  ->  ( ( M  -  1 )  < 
M  <->  ( M ... ( M  -  1
) )  =  (/) ) )
2918, 27, 28syl2anc 673 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( M  - 
1 )  <  M  <->  ( M ... ( M  -  1 ) )  =  (/) ) )
3025, 29mpbid 215 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( M ... ( M  -  1 ) )  =  (/) )
3123, 30eqtrd 2505 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M ... (
0  +  ( M  -  1 ) ) )  =  (/) )
3231adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  H  =  (/) )  ->  ( M ... ( 0  +  ( M  -  1 ) ) )  =  (/) )
33 eqcom 2478 . . . . . . . . . . . . . . 15  |-  ( H  =  (/)  <->  (/)  =  H )
3433biimpi 199 . . . . . . . . . . . . . 14  |-  ( H  =  (/)  ->  (/)  =  H )
3534adantl 473 . . . . . . . . . . . . 13  |-  ( (
ph  /\  H  =  (/) )  ->  (/)  =  H )
3617, 32, 353eqtrd 2509 . . . . . . . . . . . 12  |-  ( (
ph  /\  H  =  (/) )  ->  ( M ... N )  =  H )
3736fveq2d 5883 . . . . . . . . . . 11  |-  ( (
ph  /\  H  =  (/) )  ->  ( # `  ( M ... N ) )  =  ( # `  H
) )
3820, 19pncan3d 10008 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 1  +  ( M  -  1 ) )  =  M )
3938eqcomd 2477 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  M  =  ( 1  +  ( M  - 
1 ) ) )
4039adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  H  =  (/) )  ->  M  =  ( 1  +  ( M  -  1 ) ) )
41 1red 9676 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  H  =  (/) )  ->  1  e.  RR )
42 neqne 2651 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  H  =  (/)  ->  H  =/=  (/) )
4342adantl 473 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  -.  H  =  (/) )  ->  H  =/=  (/) )
44 fzisoeu.h . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  H  e.  Fin )
4544adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  -.  H  =  (/) )  ->  H  e.  Fin )
46 hashnncl 12585 . . . . . . . . . . . . . . . . . . . 20  |-  ( H  e.  Fin  ->  (
( # `  H )  e.  NN  <->  H  =/=  (/) ) )
4745, 46syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
( # `  H )  e.  NN  <->  H  =/=  (/) ) )
4843, 47mpbird 240 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( # `
 H )  e.  NN )
4948nnred 10646 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( # `
 H )  e.  RR )
5027zred 11063 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( M  -  1 )  e.  RR )
5150adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( M  -  1 )  e.  RR )
5248nnge1d 10674 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  H  =  (/) )  ->  1  <_  ( # `  H
) )
5341, 49, 51, 52leadd1dd 10248 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
1  +  ( M  -  1 ) )  <_  ( ( # `  H )  +  ( M  -  1 ) ) )
5453, 10syl6breqr 4436 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
1  +  ( M  -  1 ) )  <_  N )
5540, 54eqbrtrd 4416 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  H  =  (/) )  ->  M  <_  N )
5618adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  H  =  (/) )  ->  M  e.  ZZ )
57 hashcl 12576 . . . . . . . . . . . . . . . . . . 19  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
58 nn0z 10984 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  H )  e.  NN0  ->  ( # `  H
)  e.  ZZ )
5944, 57, 583syl 18 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
6059, 27zaddcld 11067 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( # `  H
)  +  ( M  -  1 ) )  e.  ZZ )
6110, 60syl5eqel 2553 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  ZZ )
6261adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  H  =  (/) )  ->  N  e.  ZZ )
63 eluz 11196 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  (
ZZ>= `  M )  <->  M  <_  N ) )
6456, 62, 63syl2anc 673 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( N  e.  ( ZZ>= `  M )  <->  M  <_  N ) )
6555, 64mpbird 240 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  H  =  (/) )  ->  N  e.  ( ZZ>= `  M )
)
66 hashfz 12640 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( # `  ( M ... N ) )  =  ( ( N  -  M )  +  1 ) )
6765, 66syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( # `
 ( M ... N ) )  =  ( ( N  -  M )  +  1 ) )
6810oveq1i 6318 . . . . . . . . . . . . . . . 16  |-  ( N  -  M )  =  ( ( ( # `  H )  +  ( M  -  1 ) )  -  M )
6944, 57syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
7069nn0cnd 10951 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( # `  H
)  e.  CC )
7170, 21, 19addsubassd 10025 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( # `  H )  +  ( M  -  1 ) )  -  M )  =  ( ( # `  H )  +  ( ( M  -  1 )  -  M ) ) )
7268, 71syl5eq 2517 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( N  -  M
)  =  ( (
# `  H )  +  ( ( M  -  1 )  -  M ) ) )
7319, 20negsubd 10011 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( M  +  -u
1 )  =  ( M  -  1 ) )
7473eqcomd 2477 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( M  -  1 )  =  ( M  +  -u 1 ) )
7574oveq1d 6323 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( M  - 
1 )  -  M
)  =  ( ( M  +  -u 1
)  -  M ) )
7620negcld 9992 . . . . . . . . . . . . . . . . . 18  |-  ( ph  -> 
-u 1  e.  CC )
7719, 76pncan2d 10007 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( M  +  -u 1 )  -  M
)  =  -u 1
)
7875, 77eqtrd 2505 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( M  - 
1 )  -  M
)  =  -u 1
)
7978oveq2d 6324 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  H
)  +  ( ( M  -  1 )  -  M ) )  =  ( ( # `  H )  +  -u
1 ) )
8072, 79eqtrd 2505 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( N  -  M
)  =  ( (
# `  H )  +  -u 1 ) )
8180oveq1d 6323 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( N  -  M )  +  1 )  =  ( ( ( # `  H
)  +  -u 1
)  +  1 ) )
8281adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
( N  -  M
)  +  1 )  =  ( ( (
# `  H )  +  -u 1 )  +  1 ) )
8370, 20negsubd 10011 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  H
)  +  -u 1
)  =  ( (
# `  H )  -  1 ) )
8483oveq1d 6323 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( # `  H )  +  -u
1 )  +  1 )  =  ( ( ( # `  H
)  -  1 )  +  1 ) )
8570, 20npcand 10009 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( # `  H )  -  1 )  +  1 )  =  ( # `  H
) )
8684, 85eqtrd 2505 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( # `  H )  +  -u
1 )  +  1 )  =  ( # `  H ) )
8786adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
( ( # `  H
)  +  -u 1
)  +  1 )  =  ( # `  H
) )
8867, 82, 873eqtrd 2509 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( # `
 ( M ... N ) )  =  ( # `  H
) )
8937, 88pm2.61dan 808 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( M ... N ) )  =  ( # `  H
) )
9089oveq2d 6324 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... ( # `
 ( M ... N ) ) )  =  ( 1 ... ( # `  H
) ) )
91 isoeq4 6231 . . . . . . . . 9  |-  ( ( 1 ... ( # `  ( M ... N
) ) )  =  ( 1 ... ( # `
 H ) )  ->  ( h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) )  <-> 
h  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  ( M ... N ) ) ) )
9290, 91syl 17 . . . . . . . 8  |-  ( ph  ->  ( h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) )  <-> 
h  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  ( M ... N ) ) ) )
9392biimpd 212 . . . . . . 7  |-  ( ph  ->  ( h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) )  ->  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) ) )
9493eximdv 1772 . . . . . 6  |-  ( ph  ->  ( E. h  h 
Isom  <  ,  <  (
( 1 ... ( # `
 ( M ... N ) ) ) ,  ( M ... N ) )  ->  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) ) )
959, 94mpi 20 . . . . 5  |-  ( ph  ->  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) )
96 fzisoeu.or . . . . . 6  |-  ( ph  ->  <  Or  H )
97 fz1iso 12666 . . . . . 6  |-  ( (  <  Or  H  /\  H  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )
9896, 44, 97syl2anc 673 . . . . 5  |-  ( ph  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )
99 eeanv 2093 . . . . 5  |-  ( E. h E. g ( h  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  H ) )  <-> 
( E. h  h 
Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  /\  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) ) )
10095, 98, 99sylanbrc 677 . . . 4  |-  ( ph  ->  E. h E. g
( h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) ) )
101 isocnv 6239 . . . . . . . 8  |-  ( h 
Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  ->  `' h  Isom  <  ,  <  ( ( M ... N ) ,  ( 1 ... ( # `  H ) ) ) )
102101ad2antrl 742 . . . . . . 7  |-  ( (
ph  /\  ( h  Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  H ) ) )  ->  `' h  Isom  <  ,  <  (
( M ... N
) ,  ( 1 ... ( # `  H
) ) ) )
103 simprr 774 . . . . . . 7  |-  ( (
ph  /\  ( h  Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  H ) ) )  ->  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )
104 isotr 6245 . . . . . . 7  |-  ( ( `' h  Isom  <  ,  <  ( ( M ... N ) ,  ( 1 ... ( # `  H ) ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )  ->  (
g  o.  `' h
)  Isom  <  ,  <  ( ( M ... N
) ,  H ) )
105102, 103, 104syl2anc 673 . . . . . 6  |-  ( (
ph  /\  ( h  Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  H ) ) )  ->  ( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
106105ex 441 . . . . 5  |-  ( ph  ->  ( ( h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )  ->  (
g  o.  `' h
)  Isom  <  ,  <  ( ( M ... N
) ,  H ) ) )
1071062eximdv 1774 . . . 4  |-  ( ph  ->  ( E. h E. g ( h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )  ->  E. h E. g ( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H ) ) )
108100, 107mpd 15 . . 3  |-  ( ph  ->  E. h E. g
( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
109 vex 3034 . . . . . . 7  |-  g  e. 
_V
110 vex 3034 . . . . . . . 8  |-  h  e. 
_V
111110cnvex 6759 . . . . . . 7  |-  `' h  e.  _V
112109, 111coex 6764 . . . . . 6  |-  ( g  o.  `' h )  e.  _V
113 isoeq1 6228 . . . . . 6  |-  ( f  =  ( g  o.  `' h )  ->  (
f  Isom  <  ,  <  ( ( M ... N
) ,  H )  <-> 
( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H ) ) )
114112, 113spcev 3127 . . . . 5  |-  ( ( g  o.  `' h
)  Isom  <  ,  <  ( ( M ... N
) ,  H )  ->  E. f  f  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
115114a1i 11 . . . 4  |-  ( ph  ->  ( ( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H )  ->  E. f 
f  Isom  <  ,  <  ( ( M ... N
) ,  H ) ) )
116115exlimdvv 1788 . . 3  |-  ( ph  ->  ( E. h E. g ( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H )  ->  E. f 
f  Isom  <  ,  <  ( ( M ... N
) ,  H ) ) )
117108, 116mpd 15 . 2  |-  ( ph  ->  E. f  f  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
118 ltwefz 12215 . . 3  |-  <  We  ( M ... N )
119 wemoiso 6797 . . 3  |-  (  < 
We  ( M ... N )  ->  E* f  f  Isom  <  ,  <  ( ( M ... N ) ,  H
) )
120118, 119mp1i 13 . 2  |-  ( ph  ->  E* f  f  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
121 eu5 2345 . 2  |-  ( E! f  f  Isom  <  ,  <  ( ( M ... N ) ,  H )  <->  ( E. f  f  Isom  <  ,  <  ( ( M ... N ) ,  H
)  /\  E* f 
f  Isom  <  ,  <  ( ( M ... N
) ,  H ) ) )
122117, 120, 121sylanbrc 677 1  |-  ( ph  ->  E! f  f  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   E!weu 2319   E*wmo 2320    =/= wne 2641    C_ wss 3390   (/)c0 3722   class class class wbr 4395    Or wor 4759    We wwe 4797   `'ccnv 4838    o. ccom 4843   ` cfv 5589    Isom wiso 5590  (class class class)co 6308   Fincfn 7587   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880   -ucneg 9881   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   #chash 12553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554
This theorem is referenced by:  fourierdlem36  38118
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