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Theorem fzisoeu 31739
Description: A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 12495 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 31706 . . . . . . . . 9  |-  ( M ... N )  C_  ZZ
2 zssre 10867 . . . . . . . . 9  |-  ZZ  C_  RR
31, 2sstri 3498 . . . . . . . 8  |-  ( M ... N )  C_  RR
4 ltso 9654 . . . . . . . 8  |-  <  Or  RR
5 soss 4807 . . . . . . . 8  |-  ( ( M ... N ) 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  ( M ... N ) ) )
63, 4, 5mp2 9 . . . . . . 7  |-  <  Or  ( M ... N )
7 fzfi 12064 . . . . . . 7  |-  ( M ... N )  e. 
Fin
8 fz1iso 12495 . . . . . . 7  |-  ( (  <  Or  ( M ... N )  /\  ( M ... N )  e.  Fin )  ->  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) ) )
96, 7, 8mp2an 670 . . . . . 6  |-  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `
 ( M ... N ) ) ) ,  ( M ... N ) )
10 fzisoeu.4 . . . . . . . . . . . . . . . 16  |-  N  =  ( ( # `  H
)  +  ( M  -  1 ) )
11 fveq2 5848 . . . . . . . . . . . . . . . . . 18  |-  ( H  =  (/)  ->  ( # `  H )  =  (
# `  (/) ) )
12 hash0 12420 . . . . . . . . . . . . . . . . . 18  |-  ( # `  (/) )  =  0
1311, 12syl6eq 2511 . . . . . . . . . . . . . . . . 17  |-  ( H  =  (/)  ->  ( # `  H )  =  0 )
1413oveq1d 6285 . . . . . . . . . . . . . . . 16  |-  ( H  =  (/)  ->  ( (
# `  H )  +  ( M  - 
1 ) )  =  ( 0  +  ( M  -  1 ) ) )
1510, 14syl5eq 2507 . . . . . . . . . . . . . . 15  |-  ( H  =  (/)  ->  N  =  ( 0  +  ( M  -  1 ) ) )
1615oveq2d 6286 . . . . . . . . . . . . . 14  |-  ( H  =  (/)  ->  ( M ... N )  =  ( M ... (
0  +  ( M  -  1 ) ) ) )
1716adantl 464 . . . . . . . . . . . . 13  |-  ( (
ph  /\  H  =  (/) )  ->  ( M ... N )  =  ( M ... ( 0  +  ( M  - 
1 ) ) ) )
18 fzisoeu.m . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  M  e.  ZZ )
1918zcnd 10966 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  M  e.  CC )
20 1cnd 9601 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  1  e.  CC )
2119, 20subcld 9922 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( M  -  1 )  e.  CC )
2221addid2d 9770 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 0  +  ( M  -  1 ) )  =  ( M  -  1 ) )
2322oveq2d 6286 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( M ... (
0  +  ( M  -  1 ) ) )  =  ( M ... ( M  - 
1 ) ) )
2418zred 10965 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  M  e.  RR )
2524ltm1d 10473 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( M  -  1 )  <  M )
26 peano2zm 10903 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
2718, 26syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
28 fzn 11705 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  ZZ  /\  ( M  -  1
)  e.  ZZ )  ->  ( ( M  -  1 )  < 
M  <->  ( M ... ( M  -  1
) )  =  (/) ) )
2918, 27, 28syl2anc 659 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( M  - 
1 )  <  M  <->  ( M ... ( M  -  1 ) )  =  (/) ) )
3025, 29mpbid 210 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( M ... ( M  -  1 ) )  =  (/) )
3123, 30eqtrd 2495 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M ... (
0  +  ( M  -  1 ) ) )  =  (/) )
3231adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  H  =  (/) )  ->  ( M ... ( 0  +  ( M  -  1 ) ) )  =  (/) )
33 eqcom 2463 . . . . . . . . . . . . . . 15  |-  ( H  =  (/)  <->  (/)  =  H )
3433biimpi 194 . . . . . . . . . . . . . 14  |-  ( H  =  (/)  ->  (/)  =  H )
3534adantl 464 . . . . . . . . . . . . 13  |-  ( (
ph  /\  H  =  (/) )  ->  (/)  =  H )
3617, 32, 353eqtrd 2499 . . . . . . . . . . . 12  |-  ( (
ph  /\  H  =  (/) )  ->  ( M ... N )  =  H )
3736fveq2d 5852 . . . . . . . . . . 11  |-  ( (
ph  /\  H  =  (/) )  ->  ( # `  ( M ... N ) )  =  ( # `  H
) )
3820, 19pncan3d 9925 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 1  +  ( M  -  1 ) )  =  M )
3938eqcomd 2462 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  M  =  ( 1  +  ( M  - 
1 ) ) )
4039adantr 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  H  =  (/) )  ->  M  =  ( 1  +  ( M  -  1 ) ) )
41 1red 9600 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  H  =  (/) )  ->  1  e.  RR )
42 neqne 31674 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  H  =  (/)  ->  H  =/=  (/) )
4342adantl 464 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  -.  H  =  (/) )  ->  H  =/=  (/) )
44 fzisoeu.h . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  H  e.  Fin )
4544adantr 463 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  -.  H  =  (/) )  ->  H  e.  Fin )
46 hashnncl 12419 . . . . . . . . . . . . . . . . . . . 20  |-  ( H  e.  Fin  ->  (
( # `  H )  e.  NN  <->  H  =/=  (/) ) )
4745, 46syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
( # `  H )  e.  NN  <->  H  =/=  (/) ) )
4843, 47mpbird 232 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( # `
 H )  e.  NN )
4948nnred 10546 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( # `
 H )  e.  RR )
5027zred 10965 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( M  -  1 )  e.  RR )
5150adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( M  -  1 )  e.  RR )
5248nnge1d 10574 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  H  =  (/) )  ->  1  <_  ( # `  H
) )
5341, 49, 51, 52leadd1dd 10162 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
1  +  ( M  -  1 ) )  <_  ( ( # `  H )  +  ( M  -  1 ) ) )
5453, 10syl6breqr 4479 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
1  +  ( M  -  1 ) )  <_  N )
5540, 54eqbrtrd 4459 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  H  =  (/) )  ->  M  <_  N )
5618adantr 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  H  =  (/) )  ->  M  e.  ZZ )
57 hashcl 12410 . . . . . . . . . . . . . . . . . . 19  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
58 nn0z 10883 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  H )  e.  NN0  ->  ( # `  H
)  e.  ZZ )
5944, 57, 583syl 20 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
6059, 27zaddcld 10969 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( # `  H
)  +  ( M  -  1 ) )  e.  ZZ )
6110, 60syl5eqel 2546 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  ZZ )
6261adantr 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  H  =  (/) )  ->  N  e.  ZZ )
63 eluz 11095 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  (
ZZ>= `  M )  <->  M  <_  N ) )
6456, 62, 63syl2anc 659 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( N  e.  ( ZZ>= `  M )  <->  M  <_  N ) )
6555, 64mpbird 232 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  H  =  (/) )  ->  N  e.  ( ZZ>= `  M )
)
66 hashfz 12469 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( # `  ( M ... N ) )  =  ( ( N  -  M )  +  1 ) )
6765, 66syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( # `
 ( M ... N ) )  =  ( ( N  -  M )  +  1 ) )
6810oveq1i 6280 . . . . . . . . . . . . . . . 16  |-  ( N  -  M )  =  ( ( ( # `  H )  +  ( M  -  1 ) )  -  M )
6944, 57syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
7069nn0cnd 10850 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( # `  H
)  e.  CC )
7170, 21, 19addsubassd 9942 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( # `  H )  +  ( M  -  1 ) )  -  M )  =  ( ( # `  H )  +  ( ( M  -  1 )  -  M ) ) )
7268, 71syl5eq 2507 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( N  -  M
)  =  ( (
# `  H )  +  ( ( M  -  1 )  -  M ) ) )
7319, 20negsubd 9928 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( M  +  -u
1 )  =  ( M  -  1 ) )
7473eqcomd 2462 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( M  -  1 )  =  ( M  +  -u 1 ) )
7574oveq1d 6285 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( M  - 
1 )  -  M
)  =  ( ( M  +  -u 1
)  -  M ) )
7620negcld 9909 . . . . . . . . . . . . . . . . . 18  |-  ( ph  -> 
-u 1  e.  CC )
7719, 76pncan2d 9924 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( M  +  -u 1 )  -  M
)  =  -u 1
)
7875, 77eqtrd 2495 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( M  - 
1 )  -  M
)  =  -u 1
)
7978oveq2d 6286 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  H
)  +  ( ( M  -  1 )  -  M ) )  =  ( ( # `  H )  +  -u
1 ) )
8072, 79eqtrd 2495 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( N  -  M
)  =  ( (
# `  H )  +  -u 1 ) )
8180oveq1d 6285 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( N  -  M )  +  1 )  =  ( ( ( # `  H
)  +  -u 1
)  +  1 ) )
8281adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
( N  -  M
)  +  1 )  =  ( ( (
# `  H )  +  -u 1 )  +  1 ) )
8370, 20negsubd 9928 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  H
)  +  -u 1
)  =  ( (
# `  H )  -  1 ) )
8483oveq1d 6285 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( # `  H )  +  -u
1 )  +  1 )  =  ( ( ( # `  H
)  -  1 )  +  1 ) )
8570, 20npcand 9926 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( # `  H )  -  1 )  +  1 )  =  ( # `  H
) )
8684, 85eqtrd 2495 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( # `  H )  +  -u
1 )  +  1 )  =  ( # `  H ) )
8786adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
( ( # `  H
)  +  -u 1
)  +  1 )  =  ( # `  H
) )
8867, 82, 873eqtrd 2499 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( # `
 ( M ... N ) )  =  ( # `  H
) )
8937, 88pm2.61dan 789 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( M ... N ) )  =  ( # `  H
) )
9089oveq2d 6286 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... ( # `
 ( M ... N ) ) )  =  ( 1 ... ( # `  H
) ) )
91 isoeq4 6193 . . . . . . . . 9  |-  ( ( 1 ... ( # `  ( M ... N
) ) )  =  ( 1 ... ( # `
 H ) )  ->  ( h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) )  <-> 
h  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  ( M ... N ) ) ) )
9290, 91syl 16 . . . . . . . 8  |-  ( ph  ->  ( h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) )  <-> 
h  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  ( M ... N ) ) ) )
9392biimpd 207 . . . . . . 7  |-  ( ph  ->  ( h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) )  ->  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) ) )
9493eximdv 1715 . . . . . 6  |-  ( ph  ->  ( E. h  h 
Isom  <  ,  <  (
( 1 ... ( # `
 ( M ... N ) ) ) ,  ( M ... N ) )  ->  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) ) )
959, 94mpi 17 . . . . 5  |-  ( ph  ->  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) )
96 fzisoeu.or . . . . . 6  |-  ( ph  ->  <  Or  H )
97 fz1iso 12495 . . . . . 6  |-  ( (  <  Or  H  /\  H  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )
9896, 44, 97syl2anc 659 . . . . 5  |-  ( ph  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )
99 eeanv 1993 . . . . 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 662 . . . 4  |-  ( ph  ->  E. h E. g
( h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) ) )
101 isocnv 6201 . . . . . . . 8  |-  ( h 
Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  ->  `' h  Isom  <  ,  <  ( ( M ... N ) ,  ( 1 ... ( # `  H ) ) ) )
102101ad2antrl 725 . . . . . . 7  |-  ( (
ph  /\  ( h  Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  H ) ) )  ->  `' h  Isom  <  ,  <  (
( M ... N
) ,  ( 1 ... ( # `  H
) ) ) )
103 simprr 755 . . . . . . 7  |-  ( (
ph  /\  ( h  Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  H ) ) )  ->  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )
104 isotr 6207 . . . . . . 7  |-  ( ( `' h  Isom  <  ,  <  ( ( M ... N ) ,  ( 1 ... ( # `  H ) ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )  ->  (
g  o.  `' h
)  Isom  <  ,  <  ( ( M ... N
) ,  H ) )
105102, 103, 104syl2anc 659 . . . . . 6  |-  ( (
ph  /\  ( h  Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  H ) ) )  ->  ( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
106105ex 432 . . . . 5  |-  ( ph  ->  ( ( h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )  ->  (
g  o.  `' h
)  Isom  <  ,  <  ( ( M ... N
) ,  H ) ) )
1071062eximdv 1717 . . . 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 3109 . . . . . . 7  |-  g  e. 
_V
110 vex 3109 . . . . . . . 8  |-  h  e. 
_V
111110cnvex 6720 . . . . . . 7  |-  `' h  e.  _V
112109, 111coex 6725 . . . . . 6  |-  ( g  o.  `' h )  e.  _V
113 isoeq1 6190 . . . . . 6  |-  ( f  =  ( g  o.  `' h )  ->  (
f  Isom  <  ,  <  ( ( M ... N
) ,  H )  <-> 
( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H ) ) )
114112, 113spcev 3198 . . . . 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 1730 . . 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 12056 . . 3  |-  <  We  ( M ... N )
119 wemoiso 6758 . . 3  |-  (  < 
We  ( M ... N )  ->  E* f  f  Isom  <  ,  <  ( ( M ... N ) ,  H
) )
120118, 119mp1i 12 . 2  |-  ( ph  ->  E* f  f  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
121 eu5 2312 . 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 662 1  |-  ( ph  ->  E! f  f  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   E!weu 2284   E*wmo 2285    =/= wne 2649    C_ wss 3461   (/)c0 3783   class class class wbr 4439    Or wor 4788    We wwe 4826   `'ccnv 4987    o. ccom 4992   ` cfv 5570    Isom wiso 5571  (class class class)co 6270   Fincfn 7509   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617    <_ cle 9618    - cmin 9796   -ucneg 9797   NNcn 10531   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12388
This theorem is referenced by:  fourierdlem36  32164
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