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Theorem erdszelem9 29994
Description: Lemma for erdsze 29997. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n  |-  ( ph  ->  N  e.  NN )
erdsze.f  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
erdszelem.i  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.j  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.t  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
Assertion
Ref Expression
erdszelem9  |-  ( ph  ->  T : ( 1 ... N ) -1-1-> ( NN  X.  NN ) )
Distinct variable groups:    x, y, n, F    n, I, x, y    n, J, x, y    n, N, x, y    ph, n, x, y
Allowed substitution hints:    T( x, y, n)

Proof of Theorem erdszelem9
Dummy variables  w  z  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 erdsze.n . . . . . 6  |-  ( ph  ->  N  e.  NN )
2 erdsze.f . . . . . 6  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
3 erdszelem.i . . . . . 6  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
4 ltso 9732 . . . . . 6  |-  <  Or  RR
51, 2, 3, 4erdszelem6 29991 . . . . 5  |-  ( ph  ->  I : ( 1 ... N ) --> NN )
65ffvelrnda 6037 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
I `  n )  e.  NN )
7 erdszelem.j . . . . . 6  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
8 gtso 9733 . . . . . 6  |-  `'  <  Or  RR
91, 2, 7, 8erdszelem6 29991 . . . . 5  |-  ( ph  ->  J : ( 1 ... N ) --> NN )
109ffvelrnda 6037 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( J `  n )  e.  NN )
11 opelxpi 4871 . . . 4  |-  ( ( ( I `  n
)  e.  NN  /\  ( J `  n )  e.  NN )  ->  <. ( I `  n
) ,  ( J `
 n ) >.  e.  ( NN  X.  NN ) )
126, 10, 11syl2anc 673 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  <. (
I `  n ) ,  ( J `  n ) >.  e.  ( NN  X.  NN ) )
13 erdszelem.t . . 3  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
1412, 13fmptd 6061 . 2  |-  ( ph  ->  T : ( 1 ... N ) --> ( NN  X.  NN ) )
15 fveq2 5879 . . . . . 6  |-  ( a  =  z  ->  ( T `  a )  =  ( T `  z ) )
16 fveq2 5879 . . . . . 6  |-  ( b  =  w  ->  ( T `  b )  =  ( T `  w ) )
1715, 16eqeqan12d 2487 . . . . 5  |-  ( ( a  =  z  /\  b  =  w )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  z
)  =  ( T `
 w ) ) )
18 eqeq12 2484 . . . . 5  |-  ( ( a  =  z  /\  b  =  w )  ->  ( a  =  b  <-> 
z  =  w ) )
1917, 18imbi12d 327 . . . 4  |-  ( ( a  =  z  /\  b  =  w )  ->  ( ( ( T `
 a )  =  ( T `  b
)  ->  a  =  b )  <->  ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) ) )
20 fveq2 5879 . . . . . . 7  |-  ( a  =  w  ->  ( T `  a )  =  ( T `  w ) )
21 fveq2 5879 . . . . . . 7  |-  ( b  =  z  ->  ( T `  b )  =  ( T `  z ) )
2220, 21eqeqan12d 2487 . . . . . 6  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  w
)  =  ( T `
 z ) ) )
23 eqcom 2478 . . . . . 6  |-  ( ( T `  w )  =  ( T `  z )  <->  ( T `  z )  =  ( T `  w ) )
2422, 23syl6bb 269 . . . . 5  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  z
)  =  ( T `
 w ) ) )
25 eqeq12 2484 . . . . . 6  |-  ( ( a  =  w  /\  b  =  z )  ->  ( a  =  b  <-> 
w  =  z ) )
26 eqcom 2478 . . . . . 6  |-  ( w  =  z  <->  z  =  w )
2725, 26syl6bb 269 . . . . 5  |-  ( ( a  =  w  /\  b  =  z )  ->  ( a  =  b  <-> 
z  =  w ) )
2824, 27imbi12d 327 . . . 4  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( ( T `
 a )  =  ( T `  b
)  ->  a  =  b )  <->  ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) ) )
29 elfzelz 11826 . . . . . . 7  |-  ( z  e.  ( 1 ... N )  ->  z  e.  ZZ )
3029zred 11063 . . . . . 6  |-  ( z  e.  ( 1 ... N )  ->  z  e.  RR )
3130ssriv 3422 . . . . 5  |-  ( 1 ... N )  C_  RR
3231a1i 11 . . . 4  |-  ( ph  ->  ( 1 ... N
)  C_  RR )
33 biidd 245 . . . 4  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
) ) )  -> 
( ( ( T `
 z )  =  ( T `  w
)  ->  z  =  w )  <->  ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) ) )
34 simpr1 1036 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  z  e.  ( 1 ... N
) )
35 fveq2 5879 . . . . . . . . . 10  |-  ( n  =  z  ->  (
I `  n )  =  ( I `  z ) )
36 fveq2 5879 . . . . . . . . . 10  |-  ( n  =  z  ->  ( J `  n )  =  ( J `  z ) )
3735, 36opeq12d 4166 . . . . . . . . 9  |-  ( n  =  z  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  z ) ,  ( J `  z ) >. )
38 opex 4664 . . . . . . . . 9  |-  <. (
I `  z ) ,  ( J `  z ) >.  e.  _V
3937, 13, 38fvmpt 5963 . . . . . . . 8  |-  ( z  e.  ( 1 ... N )  ->  ( T `  z )  =  <. ( I `  z ) ,  ( J `  z )
>. )
4034, 39syl 17 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  ( T `  z )  =  <. ( I `  z ) ,  ( J `  z )
>. )
41 simpr2 1037 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  w  e.  ( 1 ... N
) )
42 fveq2 5879 . . . . . . . . . 10  |-  ( n  =  w  ->  (
I `  n )  =  ( I `  w ) )
43 fveq2 5879 . . . . . . . . . 10  |-  ( n  =  w  ->  ( J `  n )  =  ( J `  w ) )
4442, 43opeq12d 4166 . . . . . . . . 9  |-  ( n  =  w  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  w ) ,  ( J `  w ) >. )
45 opex 4664 . . . . . . . . 9  |-  <. (
I `  w ) ,  ( J `  w ) >.  e.  _V
4644, 13, 45fvmpt 5963 . . . . . . . 8  |-  ( w  e.  ( 1 ... N )  ->  ( T `  w )  =  <. ( I `  w ) ,  ( J `  w )
>. )
4741, 46syl 17 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  ( T `  w )  =  <. ( I `  w ) ,  ( J `  w )
>. )
4840, 47eqeq12d 2486 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( T `  z
)  =  ( T `
 w )  <->  <. ( I `
 z ) ,  ( J `  z
) >.  =  <. (
I `  w ) ,  ( J `  w ) >. )
)
49 fvex 5889 . . . . . . . 8  |-  ( I `
 z )  e. 
_V
50 fvex 5889 . . . . . . . 8  |-  ( J `
 z )  e. 
_V
5149, 50opth 4676 . . . . . . 7  |-  ( <.
( I `  z
) ,  ( J `
 z ) >.  =  <. ( I `  w ) ,  ( J `  w )
>. 
<->  ( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) ) )
5234, 30syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  z  e.  RR )
5331, 41sseldi 3416 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  w  e.  RR )
54 simpr3 1038 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  z  <_  w )
5552, 53, 54leltned 9805 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
z  <  w  <->  w  =/=  z ) )
562adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  F : ( 1 ... N ) -1-1-> RR )
57 f1fveq 6181 . . . . . . . . . . . . . . . . 17  |-  ( ( F : ( 1 ... N ) -1-1-> RR  /\  ( z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N ) ) )  ->  (
( F `  z
)  =  ( F `
 w )  <->  z  =  w ) )
5856, 34, 41, 57syl12anc 1290 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( F `  z
)  =  ( F `
 w )  <->  z  =  w ) )
5958, 26syl6bbr 271 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( F `  z
)  =  ( F `
 w )  <->  w  =  z ) )
6059necon3bid 2687 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( F `  z
)  =/=  ( F `
 w )  <->  w  =/=  z ) )
6155, 60bitr4d 264 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
z  <  w  <->  ( F `  z )  =/=  ( F `  w )
) )
6261biimpa 492 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  z )  =/=  ( F `  w )
)
63 f1f 5792 . . . . . . . . . . . . . . . 16  |-  ( F : ( 1 ... N ) -1-1-> RR  ->  F : ( 1 ... N ) --> RR )
642, 63syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( 1 ... N ) --> RR )
6564ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  F :
( 1 ... N
) --> RR )
6634adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  z  e.  ( 1 ... N
) )
6765, 66ffvelrnd 6038 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  z )  e.  RR )
6841adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  w  e.  ( 1 ... N
) )
6965, 68ffvelrnd 6038 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  w )  e.  RR )
7067, 69lttri2d 9791 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( ( F `  z )  =/=  ( F `  w
)  <->  ( ( F `
 z )  < 
( F `  w
)  \/  ( F `
 w )  < 
( F `  z
) ) ) )
7162, 70mpbid 215 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( ( F `  z )  <  ( F `  w
)  \/  ( F `
 w )  < 
( F `  z
) ) )
721ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  N  e.  NN )
732ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  F :
( 1 ... N
) -1-1-> RR )
74 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  z  <  w )
7572, 73, 3, 4, 66, 68, 74erdszelem8 29993 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( (
I `  z )  =  ( I `  w )  ->  -.  ( F `  z )  <  ( F `  w ) ) )
7672, 73, 7, 8, 66, 68, 74erdszelem8 29993 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( ( J `  z )  =  ( J `  w )  ->  -.  ( F `  z ) `'  <  ( F `  w ) ) )
7775, 76anim12d 572 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( (
( I `  z
)  =  ( I `
 w )  /\  ( J `  z )  =  ( J `  w ) )  -> 
( -.  ( F `
 z )  < 
( F `  w
)  /\  -.  ( F `  z ) `'  <  ( F `  w ) ) ) )
78 ioran 498 . . . . . . . . . . . . 13  |-  ( -.  ( ( F `  z )  <  ( F `  w )  \/  ( F `  w
)  <  ( F `  z ) )  <->  ( -.  ( F `  z )  <  ( F `  w )  /\  -.  ( F `  w )  <  ( F `  z ) ) )
79 fvex 5889 . . . . . . . . . . . . . . . 16  |-  ( F `
 z )  e. 
_V
80 fvex 5889 . . . . . . . . . . . . . . . 16  |-  ( F `
 w )  e. 
_V
8179, 80brcnv 5022 . . . . . . . . . . . . . . 15  |-  ( ( F `  z ) `'  <  ( F `  w )  <->  ( F `  w )  <  ( F `  z )
)
8281notbii 303 . . . . . . . . . . . . . 14  |-  ( -.  ( F `  z
) `'  <  ( F `  w )  <->  -.  ( F `  w
)  <  ( F `  z ) )
8382anbi2i 708 . . . . . . . . . . . . 13  |-  ( ( -.  ( F `  z )  <  ( F `  w )  /\  -.  ( F `  z ) `'  <  ( F `  w ) )  <->  ( -.  ( F `  z )  <  ( F `  w
)  /\  -.  ( F `  w )  <  ( F `  z
) ) )
8478, 83bitr4i 260 . . . . . . . . . . . 12  |-  ( -.  ( ( F `  z )  <  ( F `  w )  \/  ( F `  w
)  <  ( F `  z ) )  <->  ( -.  ( F `  z )  <  ( F `  w )  /\  -.  ( F `  z ) `'  <  ( F `  w ) ) )
8577, 84syl6ibr 235 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( (
( I `  z
)  =  ( I `
 w )  /\  ( J `  z )  =  ( J `  w ) )  ->  -.  ( ( F `  z )  <  ( F `  w )  \/  ( F `  w
)  <  ( F `  z ) ) ) )
8671, 85mt2d 121 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  -.  (
( I `  z
)  =  ( I `
 w )  /\  ( J `  z )  =  ( J `  w ) ) )
8786ex 441 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
z  <  w  ->  -.  ( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) ) ) )
8855, 87sylbird 243 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
w  =/=  z  ->  -.  ( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) ) ) )
8988necon4ad 2662 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) )  ->  w  =  z ) )
9051, 89syl5bi 225 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  ( <. ( I `  z
) ,  ( J `
 z ) >.  =  <. ( I `  w ) ,  ( J `  w )
>.  ->  w  =  z ) )
9148, 90sylbid 223 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( T `  z
)  =  ( T `
 w )  ->  w  =  z )
)
9291, 26syl6ib 234 . . . 4  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( T `  z
)  =  ( T `
 w )  -> 
z  =  w ) )
9319, 28, 32, 33, 92wlogle 10168 . . 3  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
) ) )  -> 
( ( T `  z )  =  ( T `  w )  ->  z  =  w ) )
9493ralrimivva 2814 . 2  |-  ( ph  ->  A. z  e.  ( 1 ... N ) A. w  e.  ( 1 ... N ) ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) )
95 dff13 6177 . 2  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  <->  ( T : ( 1 ... N ) --> ( NN 
X.  NN )  /\  A. z  e.  ( 1 ... N ) A. w  e.  ( 1 ... N ) ( ( T `  z
)  =  ( T `
 w )  -> 
z  =  w ) ) )
9614, 94, 95sylanbrc 677 1  |-  ( ph  ->  T : ( 1 ... N ) -1-1-> ( NN  X.  NN ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   {crab 2760    C_ wss 3390   ~Pcpw 3942   <.cop 3965   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838    |` cres 4841   "cima 4842   -->wf 5585   -1-1->wf1 5586   ` cfv 5589    Isom wiso 5590  (class class class)co 6308   supcsup 7972   RRcr 9556   1c1 9558    < clt 9693    <_ cle 9694   NNcn 10631   ...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-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  ax-pre-sup 9635
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-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-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-card 8391  df-cda 8616  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:  erdszelem10  29995
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