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Theorem fzisoeu 31400
Description: A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 12492 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 fzisoeu.or . . . . . . . . 9  |-  ( ph  ->  <  Or  H )
2 fzisoeu.h . . . . . . . . 9  |-  ( ph  ->  H  e.  Fin )
3 fz1iso 12492 . . . . . . . . 9  |-  ( (  <  Or  H  /\  H  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )
41, 2, 3syl2anc 661 . . . . . . . 8  |-  ( ph  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )
5 fzssz 31366 . . . . . . . . . . . . 13  |-  ( M ... N )  C_  ZZ
6 zssre 10883 . . . . . . . . . . . . 13  |-  ZZ  C_  RR
75, 6sstri 3518 . . . . . . . . . . . 12  |-  ( M ... N )  C_  RR
8 ltso 9677 . . . . . . . . . . . 12  |-  <  Or  RR
9 soss 4824 . . . . . . . . . . . 12  |-  ( ( M ... N ) 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  ( M ... N ) ) )
107, 8, 9mp2 9 . . . . . . . . . . 11  |-  <  Or  ( M ... N )
11 fzfi 12062 . . . . . . . . . . 11  |-  ( M ... N )  e. 
Fin
12 fz1iso 12492 . . . . . . . . . . 11  |-  ( (  <  Or  ( M ... N )  /\  ( M ... N )  e.  Fin )  ->  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) ) )
1310, 11, 12mp2an 672 . . . . . . . . . 10  |-  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `
 ( M ... N ) ) ) ,  ( M ... N ) )
1413a1i 11 . . . . . . . . 9  |-  ( ph  ->  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) ) )
15 fzisoeu.4 . . . . . . . . . . . . . . . . . . . 20  |-  N  =  ( ( # `  H
)  +  ( M  -  1 ) )
1615a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( H  =  (/)  ->  N  =  ( ( # `  H
)  +  ( M  -  1 ) ) )
17 fveq2 5872 . . . . . . . . . . . . . . . . . . . . 21  |-  ( H  =  (/)  ->  ( # `  H )  =  (
# `  (/) ) )
18 hash0 12417 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( # `  (/) )  =  0
1918a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( H  =  (/)  ->  ( # `  (/) )  =  0 )
2017, 19eqtrd 2508 . . . . . . . . . . . . . . . . . . . 20  |-  ( H  =  (/)  ->  ( # `  H )  =  0 )
2120oveq1d 6310 . . . . . . . . . . . . . . . . . . 19  |-  ( H  =  (/)  ->  ( (
# `  H )  +  ( M  - 
1 ) )  =  ( 0  +  ( M  -  1 ) ) )
2216, 21eqtrd 2508 . . . . . . . . . . . . . . . . . 18  |-  ( H  =  (/)  ->  N  =  ( 0  +  ( M  -  1 ) ) )
2322oveq2d 6311 . . . . . . . . . . . . . . . . 17  |-  ( H  =  (/)  ->  ( M ... N )  =  ( M ... (
0  +  ( M  -  1 ) ) ) )
2423adantl 466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  H  =  (/) )  ->  ( M ... N )  =  ( M ... ( 0  +  ( M  - 
1 ) ) ) )
25 fzisoeu.m . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  M  e.  ZZ )
2625zcnd 10979 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  M  e.  CC )
27 ax-1cn 9562 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  e.  CC
2827a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  1  e.  CC )
2926, 28subcld 9942 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( M  -  1 )  e.  CC )
3029addid2d 9792 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( 0  +  ( M  -  1 ) )  =  ( M  -  1 ) )
3130oveq2d 6311 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( M ... (
0  +  ( M  -  1 ) ) )  =  ( M ... ( M  - 
1 ) ) )
326, 25sseldi 3507 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  M  e.  RR )
3332ltm1d 10490 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( M  -  1 )  <  M )
34 peano2zm 10918 . . . . . . . . . . . . . . . . . . . . 21  |-  ( M  e.  ZZ  ->  ( M  -  1 )  e.  ZZ )
3525, 34syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( M  -  1 )  e.  ZZ )
36 fzn 11714 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( M  e.  ZZ  /\  ( M  -  1
)  e.  ZZ )  ->  ( ( M  -  1 )  < 
M  <->  ( M ... ( M  -  1
) )  =  (/) ) )
3725, 35, 36syl2anc 661 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( M  - 
1 )  <  M  <->  ( M ... ( M  -  1 ) )  =  (/) ) )
3833, 37mpbid 210 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( M ... ( M  -  1 ) )  =  (/) )
3931, 38eqtrd 2508 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( M ... (
0  +  ( M  -  1 ) ) )  =  (/) )
4039adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  H  =  (/) )  ->  ( M ... ( 0  +  ( M  -  1 ) ) )  =  (/) )
41 eqcom 2476 . . . . . . . . . . . . . . . . . 18  |-  ( H  =  (/)  <->  (/)  =  H )
4241biimpi 194 . . . . . . . . . . . . . . . . 17  |-  ( H  =  (/)  ->  (/)  =  H )
4342adantl 466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  H  =  (/) )  ->  (/)  =  H )
4424, 40, 433eqtrd 2512 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  H  =  (/) )  ->  ( M ... N )  =  H )
4544fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  H  =  (/) )  ->  ( # `  ( M ... N ) )  =  ( # `  H
) )
4628, 26pncan3d 9945 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( 1  +  ( M  -  1 ) )  =  M )
4746eqcomd 2475 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  M  =  ( 1  +  ( M  - 
1 ) ) )
4847adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  -.  H  =  (/) )  ->  M  =  ( 1  +  ( M  -  1 ) ) )
49 1re 9607 . . . . . . . . . . . . . . . . . . . . 21  |-  1  e.  RR
5049a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  -.  H  =  (/) )  ->  1  e.  RR )
51 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  H  =  (/)  ->  -.  H  =  (/) )
5251neqned 2670 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( -.  H  =  (/)  ->  H  =/=  (/) )
5352adantl 466 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  -.  H  =  (/) )  ->  H  =/=  (/) )
542adantr 465 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
ph  /\  -.  H  =  (/) )  ->  H  e.  Fin )
55 hashnncl 12416 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( H  e.  Fin  ->  (
( # `  H )  e.  NN  <->  H  =/=  (/) ) )
5654, 55syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
( # `  H )  e.  NN  <->  H  =/=  (/) ) )
5753, 56mpbird 232 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( # `
 H )  e.  NN )
5857nnred 10563 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( # `
 H )  e.  RR )
596, 35sseldi 3507 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( M  -  1 )  e.  RR )
6059adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( M  -  1 )  e.  RR )
6157nnge1d 10590 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  -.  H  =  (/) )  ->  1  <_  ( # `  H
) )
6250, 58, 60, 61leadd1dd 10178 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
1  +  ( M  -  1 ) )  <_  ( ( # `  H )  +  ( M  -  1 ) ) )
6315eqcomi 2480 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  H )  +  ( M  - 
1 ) )  =  N
6463a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
( # `  H )  +  ( M  - 
1 ) )  =  N )
6562, 64breqtrd 4477 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
1  +  ( M  -  1 ) )  <_  N )
6648, 65eqbrtrd 4473 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  H  =  (/) )  ->  M  <_  N )
6725adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  -.  H  =  (/) )  ->  M  e.  ZZ )
68 hashcl 12408 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( H  e.  Fin  ->  ( # `
 H )  e. 
NN0 )
69 nn0z 10899 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  H )  e.  NN0  ->  ( # `  H
)  e.  ZZ )
702, 68, 693syl 20 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( # `  H
)  e.  ZZ )
7170, 35zaddcld 10982 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( # `  H
)  +  ( M  -  1 ) )  e.  ZZ )
7215, 71syl5eqel 2559 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  ZZ )
7372adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  -.  H  =  (/) )  ->  N  e.  ZZ )
74 eluz 11107 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  (
ZZ>= `  M )  <->  M  <_  N ) )
7567, 73, 74syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( N  e.  ( ZZ>= `  M )  <->  M  <_  N ) )
7666, 75mpbird 232 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  -.  H  =  (/) )  ->  N  e.  ( ZZ>= `  M )
)
77 hashfz 12465 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( # `  ( M ... N ) )  =  ( ( N  -  M )  +  1 ) )
7876, 77syl 16 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( # `
 ( M ... N ) )  =  ( ( N  -  M )  +  1 ) )
7915oveq1i 6305 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  -  M )  =  ( ( ( # `  H )  +  ( M  -  1 ) )  -  M )
8079a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( N  -  M
)  =  ( ( ( # `  H
)  +  ( M  -  1 ) )  -  M ) )
812, 68syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( # `  H
)  e.  NN0 )
82 nn0cn 10817 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  H )  e.  NN0  ->  ( # `  H
)  e.  CC )
8381, 82syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( # `  H
)  e.  CC )
8483, 29, 26addsubassd 9962 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( ( # `  H )  +  ( M  -  1 ) )  -  M )  =  ( ( # `  H )  +  ( ( M  -  1 )  -  M ) ) )
8580, 84eqtrd 2508 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( N  -  M
)  =  ( (
# `  H )  +  ( ( M  -  1 )  -  M ) ) )
8626, 28negsubd 9948 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( M  +  -u
1 )  =  ( M  -  1 ) )
8786eqcomd 2475 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( M  -  1 )  =  ( M  +  -u 1 ) )
8887oveq1d 6310 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( M  - 
1 )  -  M
)  =  ( ( M  +  -u 1
)  -  M ) )
8928negcld 9929 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  -> 
-u 1  e.  CC )
9026, 89pncan2d 9944 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( M  +  -u 1 )  -  M
)  =  -u 1
)
9188, 90eqtrd 2508 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( M  - 
1 )  -  M
)  =  -u 1
)
9291oveq2d 6311 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( # `  H
)  +  ( ( M  -  1 )  -  M ) )  =  ( ( # `  H )  +  -u
1 ) )
9385, 92eqtrd 2508 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( N  -  M
)  =  ( (
# `  H )  +  -u 1 ) )
9493oveq1d 6310 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( N  -  M )  +  1 )  =  ( ( ( # `  H
)  +  -u 1
)  +  1 ) )
9594adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
( N  -  M
)  +  1 )  =  ( ( (
# `  H )  +  -u 1 )  +  1 ) )
9683, 28negsubd 9948 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( # `  H
)  +  -u 1
)  =  ( (
# `  H )  -  1 ) )
9796oveq1d 6310 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( # `  H )  +  -u
1 )  +  1 )  =  ( ( ( # `  H
)  -  1 )  +  1 ) )
9883, 28npcand 9946 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( # `  H )  -  1 )  +  1 )  =  ( # `  H
) )
9997, 98eqtrd 2508 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( # `  H )  +  -u
1 )  +  1 )  =  ( # `  H ) )
10099adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  H  =  (/) )  ->  (
( ( # `  H
)  +  -u 1
)  +  1 )  =  ( # `  H
) )
10178, 95, 1003eqtrd 2512 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  H  =  (/) )  ->  ( # `
 ( M ... N ) )  =  ( # `  H
) )
10245, 101pm2.61dan 789 . . . . . . . . . . . . 13  |-  ( ph  ->  ( # `  ( M ... N ) )  =  ( # `  H
) )
103102oveq2d 6311 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1 ... ( # `
 ( M ... N ) ) )  =  ( 1 ... ( # `  H
) ) )
104 isoeq4 6217 . . . . . . . . . . . 12  |-  ( ( 1 ... ( # `  ( M ... N
) ) )  =  ( 1 ... ( # `
 H ) )  ->  ( h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) )  <-> 
h  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  ( M ... N ) ) ) )
105103, 104syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) )  <-> 
h  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  ( M ... N ) ) ) )
106105biimpd 207 . . . . . . . . . 10  |-  ( ph  ->  ( h  Isom  <  ,  <  ( ( 1 ... ( # `  ( M ... N ) ) ) ,  ( M ... N ) )  ->  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) ) )
107106eximdv 1686 . . . . . . . . 9  |-  ( ph  ->  ( E. h  h 
Isom  <  ,  <  (
( 1 ... ( # `
 ( M ... N ) ) ) ,  ( M ... N ) )  ->  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) ) )
10814, 107mpd 15 . . . . . . . 8  |-  ( ph  ->  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) )
1094, 108jca 532 . . . . . . 7  |-  ( ph  ->  ( E. g  g 
Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  H )  /\  E. h  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) ) )
110109ancomd 451 . . . . . 6  |-  ( ph  ->  ( E. h  h 
Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  /\  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) ) )
111 nfv 1683 . . . . . . 7  |-  F/ g  h  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  ( M ... N ) )
112 nfv 1683 . . . . . . 7  |-  F/ h  g  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  H )
113111, 112eean 1956 . . . . . 6  |-  ( 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
) ) )
114110, 113sylibr 212 . . . . 5  |-  ( ph  ->  E. h E. g
( h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) ) )
115 isocnv 6225 . . . . . . . . . . 11  |-  ( h 
Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  ->  `' h  Isom  <  ,  <  ( ( M ... N ) ,  ( 1 ... ( # `  H ) ) ) )
116115adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) )  ->  `' h  Isom  <  ,  <  ( ( M ... N
) ,  ( 1 ... ( # `  H
) ) ) )
117 eqidd 2468 . . . . . . . . . . 11  |-  ( (
ph  /\  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) )  ->  (
1 ... ( # `  H
) )  =  ( 1 ... ( # `  H ) ) )
118 isoeq5 6218 . . . . . . . . . . 11  |-  ( ( 1 ... ( # `  H ) )  =  ( 1 ... ( # `
 H ) )  ->  ( `' h  Isom  <  ,  <  (
( M ... N
) ,  ( 1 ... ( # `  H
) ) )  <->  `' h  Isom  <  ,  <  (
( M ... N
) ,  ( 1 ... ( # `  H
) ) ) ) )
119117, 118syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) )  ->  ( `' h  Isom  <  ,  <  ( ( M ... N ) ,  ( 1 ... ( # `  H ) ) )  <->  `' h  Isom  <  ,  <  ( ( M ... N ) ,  ( 1 ... ( # `  H ) ) ) ) )
120116, 119mpbid 210 . . . . . . . . 9  |-  ( (
ph  /\  h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) ) )  ->  `' h  Isom  <  ,  <  ( ( M ... N
) ,  ( 1 ... ( # `  H
) ) ) )
121120adantrr 716 . . . . . . . 8  |-  ( (
ph  /\  ( h  Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  H ) ) )  ->  `' h  Isom  <  ,  <  (
( M ... N
) ,  ( 1 ... ( # `  H
) ) ) )
122 simprr 756 . . . . . . . 8  |-  ( (
ph  /\  ( h  Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  H ) ) )  ->  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )
123 isotr 6231 . . . . . . . 8  |-  ( ( `' h  Isom  <  ,  <  ( ( M ... N ) ,  ( 1 ... ( # `  H ) ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )  ->  (
g  o.  `' h
)  Isom  <  ,  <  ( ( M ... N
) ,  H ) )
124121, 122, 123syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  ( h  Isom  <  ,  <  (
( 1 ... ( # `
 H ) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `
 H ) ) ,  H ) ) )  ->  ( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
125124ex 434 . . . . . 6  |-  ( ph  ->  ( ( h  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  ( M ... N ) )  /\  g  Isom  <  ,  <  ( ( 1 ... ( # `  H
) ) ,  H
) )  ->  (
g  o.  `' h
)  Isom  <  ,  <  ( ( M ... N
) ,  H ) ) )
1261252eximdv 1688 . . . . 5  |-  ( 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 ) ) )
127114, 126mpd 15 . . . 4  |-  ( ph  ->  E. h E. g
( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
128 vex 3121 . . . . . . . 8  |-  g  e. 
_V
129 vex 3121 . . . . . . . . 9  |-  h  e. 
_V
130129cnvex 6742 . . . . . . . 8  |-  `' h  e.  _V
131128, 130coex 6747 . . . . . . 7  |-  ( g  o.  `' h )  e.  _V
132 isoeq1 6214 . . . . . . 7  |-  ( f  =  ( g  o.  `' h )  ->  (
f  Isom  <  ,  <  ( ( M ... N
) ,  H )  <-> 
( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H ) ) )
133131, 132spcev 3210 . . . . . 6  |-  ( ( g  o.  `' h
)  Isom  <  ,  <  ( ( M ... N
) ,  H )  ->  E. f  f  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
134133a1i 11 . . . . 5  |-  ( ph  ->  ( ( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H )  ->  E. f 
f  Isom  <  ,  <  ( ( M ... N
) ,  H ) ) )
135134exlimdvv 1701 . . . 4  |-  ( ph  ->  ( E. h E. g ( g  o.  `' h )  Isom  <  ,  <  ( ( M ... N ) ,  H )  ->  E. f 
f  Isom  <  ,  <  ( ( M ... N
) ,  H ) ) )
136127, 135mpd 15 . . 3  |-  ( ph  ->  E. f  f  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
137 ltwefz 12054 . . . . 5  |-  <  We  ( M ... N )
138137a1i 11 . . . 4  |-  ( ph  ->  <  We  ( M ... N ) )
139 wemoiso 6780 . . . 4  |-  (  < 
We  ( M ... N )  ->  E* f  f  Isom  <  ,  <  ( ( M ... N ) ,  H
) )
140138, 139syl 16 . . 3  |-  ( ph  ->  E* f  f  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
141136, 140jca 532 . 2  |-  ( ph  ->  ( E. f  f 
Isom  <  ,  <  (
( M ... N
) ,  H )  /\  E* f  f 
Isom  <  ,  <  (
( M ... N
) ,  H ) ) )
142 eu5 2305 . 2  |-  ( E! f  f  Isom  <  ,  <  ( ( M ... N ) ,  H )  <->  ( E. f  f  Isom  <  ,  <  ( ( M ... N ) ,  H
)  /\  E* f 
f  Isom  <  ,  <  ( ( M ... N
) ,  H ) ) )
143141, 142sylibr 212 1  |-  ( ph  ->  E! f  f  Isom  <  ,  <  ( ( M ... N ) ,  H ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275   E*wmo 2276    =/= wne 2662    C_ wss 3481   (/)c0 3790   class class class wbr 4453    Or wor 4805    We wwe 4843   `'ccnv 5004    o. ccom 5009   ` cfv 5594    Isom wiso 5595  (class class class)co 6295   Fincfn 7528   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    < clt 9640    <_ cle 9641    - cmin 9817   -ucneg 9818   NNcn 10548   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094   ...cfz 11684   #chash 12385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-hash 12386
This theorem is referenced by:  fourierdlem36  31766  fourierdlem54  31784
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