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Theorem erdsze2lem1 23749
Description: Lemma for erdsze2 23751. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze2.r  |-  ( ph  ->  R  e.  NN )
erdsze2.s  |-  ( ph  ->  S  e.  NN )
erdsze2.f  |-  ( ph  ->  F : A -1-1-> RR )
erdsze2.a  |-  ( ph  ->  A  C_  RR )
erdsze2lem.n  |-  N  =  ( ( R  - 
1 )  x.  ( S  -  1 ) )
erdsze2lem.l  |-  ( ph  ->  N  <  ( # `  A ) )
Assertion
Ref Expression
erdsze2lem1  |-  ( ph  ->  E. f ( f : ( 1 ... ( N  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  +  1 ) ) ,  ran  f
) ) )
Distinct variable groups:    A, f    f, F    R, f    S, f   
f, N    ph, f

Proof of Theorem erdsze2lem1
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 erdsze2lem.n . . . . . . . . 9  |-  N  =  ( ( R  - 
1 )  x.  ( S  -  1 ) )
2 erdsze2.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  NN )
3 nnm1nn0 10021 . . . . . . . . . . 11  |-  ( R  e.  NN  ->  ( R  -  1 )  e.  NN0 )
42, 3syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( R  -  1 )  e.  NN0 )
5 erdsze2.s . . . . . . . . . . 11  |-  ( ph  ->  S  e.  NN )
6 nnm1nn0 10021 . . . . . . . . . . 11  |-  ( S  e.  NN  ->  ( S  -  1 )  e.  NN0 )
75, 6syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( S  -  1 )  e.  NN0 )
84, 7nn0mulcld 10039 . . . . . . . . 9  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  e.  NN0 )
91, 8syl5eqel 2380 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
10 peano2nn0 10020 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
11 hashfz1 11361 . . . . . . . 8  |-  ( ( N  +  1 )  e.  NN0  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
129, 10, 113syl 18 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( N  + 
1 ) )
1312adantr 451 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
14 erdsze2lem.l . . . . . . . 8  |-  ( ph  ->  N  <  ( # `  A ) )
1514adantr 451 . . . . . . 7  |-  ( (
ph  /\  A  e.  Fin )  ->  N  < 
( # `  A ) )
16 hashcl 11366 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
17 nn0ltp1le 10090 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( # `  A )  e.  NN0 )  -> 
( N  <  ( # `
 A )  <->  ( N  +  1 )  <_ 
( # `  A ) ) )
189, 16, 17syl2an 463 . . . . . . 7  |-  ( (
ph  /\  A  e.  Fin )  ->  ( N  <  ( # `  A
)  <->  ( N  + 
1 )  <_  ( # `
 A ) ) )
1915, 18mpbid 201 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( N  +  1 )  <_ 
( # `  A ) )
2013, 19eqbrtrd 4059 . . . . 5  |-  ( (
ph  /\  A  e.  Fin )  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  <_  ( # `
 A ) )
21 fzfid 11051 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( 1 ... ( N  + 
1 ) )  e. 
Fin )
22 simpr 447 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  A  e. 
Fin )
23 hashdom 11377 . . . . . 6  |-  ( ( ( 1 ... ( N  +  1 ) )  e.  Fin  /\  A  e.  Fin )  ->  ( ( # `  (
1 ... ( N  + 
1 ) ) )  <_  ( # `  A
)  <->  ( 1 ... ( N  +  1 ) )  ~<_  A ) )
2421, 22, 23syl2anc 642 . . . . 5  |-  ( (
ph  /\  A  e.  Fin )  ->  ( (
# `  ( 1 ... ( N  +  1 ) ) )  <_ 
( # `  A )  <-> 
( 1 ... ( N  +  1 ) )  ~<_  A ) )
2520, 24mpbid 201 . . . 4  |-  ( (
ph  /\  A  e.  Fin )  ->  ( 1 ... ( N  + 
1 ) )  ~<_  A )
26 simpr 447 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  -.  A  e.  Fin )
27 fzfid 11051 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  (
1 ... ( N  + 
1 ) )  e. 
Fin )
28 isinffi 7641 . . . . . 6  |-  ( ( -.  A  e.  Fin  /\  ( 1 ... ( N  +  1 ) )  e.  Fin )  ->  E. f  f : ( 1 ... ( N  +  1 ) ) -1-1-> A )
2926, 27, 28syl2anc 642 . . . . 5  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  E. f 
f : ( 1 ... ( N  + 
1 ) ) -1-1-> A
)
30 erdsze2.a . . . . . . . 8  |-  ( ph  ->  A  C_  RR )
31 reex 8844 . . . . . . . 8  |-  RR  e.  _V
32 ssexg 4176 . . . . . . . 8  |-  ( ( A  C_  RR  /\  RR  e.  _V )  ->  A  e.  _V )
3330, 31, 32sylancl 643 . . . . . . 7  |-  ( ph  ->  A  e.  _V )
3433adantr 451 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  A  e.  _V )
35 brdomg 6888 . . . . . 6  |-  ( A  e.  _V  ->  (
( 1 ... ( N  +  1 ) )  ~<_  A  <->  E. f 
f : ( 1 ... ( N  + 
1 ) ) -1-1-> A
) )
3634, 35syl 15 . . . . 5  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  (
( 1 ... ( N  +  1 ) )  ~<_  A  <->  E. f 
f : ( 1 ... ( N  + 
1 ) ) -1-1-> A
) )
3729, 36mpbird 223 . . . 4  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  (
1 ... ( N  + 
1 ) )  ~<_  A )
3825, 37pm2.61dan 766 . . 3  |-  ( ph  ->  ( 1 ... ( N  +  1 ) )  ~<_  A )
39 domeng 6892 . . . 4  |-  ( A  e.  _V  ->  (
( 1 ... ( N  +  1 ) )  ~<_  A  <->  E. s
( ( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) ) )
4033, 39syl 15 . . 3  |-  ( ph  ->  ( ( 1 ... ( N  +  1 ) )  ~<_  A  <->  E. s
( ( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) ) )
4138, 40mpbid 201 . 2  |-  ( ph  ->  E. s ( ( 1 ... ( N  +  1 ) ) 
~~  s  /\  s  C_  A ) )
42 simprr 733 . . . . . . . 8  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  s  C_  A )
4330adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  A  C_  RR )
4442, 43sstrd 3202 . . . . . . 7  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  s  C_  RR )
45 ltso 8919 . . . . . . 7  |-  <  Or  RR
46 soss 4348 . . . . . . 7  |-  ( s 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  s ) )
4744, 45, 46ee10 1366 . . . . . 6  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  <  Or  s )
48 fzfid 11051 . . . . . . 7  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
1 ... ( N  + 
1 ) )  e. 
Fin )
49 simprl 732 . . . . . . . 8  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
1 ... ( N  + 
1 ) )  ~~  s )
50 enfi 7095 . . . . . . . 8  |-  ( ( 1 ... ( N  +  1 ) ) 
~~  s  ->  (
( 1 ... ( N  +  1 ) )  e.  Fin  <->  s  e.  Fin ) )
5149, 50syl 15 . . . . . . 7  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
( 1 ... ( N  +  1 ) )  e.  Fin  <->  s  e.  Fin ) )
5248, 51mpbid 201 . . . . . 6  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  s  e.  Fin )
53 fz1iso 11416 . . . . . 6  |-  ( (  <  Or  s  /\  s  e.  Fin )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )
5447, 52, 53syl2anc 642 . . . . 5  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( # `
 s ) ) ,  s ) )
55 isof1o 5838 . . . . . . . . . . . 12  |-  ( f 
Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  s )  -> 
f : ( 1 ... ( # `  s
) ) -1-1-onto-> s )
5655adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( # `  s
) ) -1-1-onto-> s )
57 hashen 11362 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1 ... ( N  +  1 ) )  e.  Fin  /\  s  e.  Fin )  ->  ( ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( # `  s
)  <->  ( 1 ... ( N  +  1 ) )  ~~  s
) )
5848, 52, 57syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
( # `  ( 1 ... ( N  + 
1 ) ) )  =  ( # `  s
)  <->  ( 1 ... ( N  +  1 ) )  ~~  s
) )
5949, 58mpbird 223 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( # `
 ( 1 ... ( N  +  1 ) ) )  =  ( # `  s
) )
6012adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( # `
 ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
6159, 60eqtr3d 2330 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( # `
 s )  =  ( N  +  1 ) )
6261adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  ( # `
 s )  =  ( N  +  1 ) )
6362oveq2d 5890 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
1 ... ( # `  s
) )  =  ( 1 ... ( N  +  1 ) ) )
64 f1oeq2 5480 . . . . . . . . . . . 12  |-  ( ( 1 ... ( # `  s ) )  =  ( 1 ... ( N  +  1 ) )  ->  ( f : ( 1 ... ( # `  s
) ) -1-1-onto-> s  <->  f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> s ) )
6563, 64syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
f : ( 1 ... ( # `  s
) ) -1-1-onto-> s  <->  f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> s ) )
6656, 65mpbid 201 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> s )
67 f1of1 5487 . . . . . . . . . 10  |-  ( f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> s  ->  f : ( 1 ... ( N  +  1 ) ) -1-1-> s )
6866, 67syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( N  +  1 ) ) -1-1-> s )
69 simplrr 737 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  s  C_  A )
70 f1ss 5458 . . . . . . . . 9  |-  ( ( f : ( 1 ... ( N  + 
1 ) ) -1-1-> s  /\  s  C_  A
)  ->  f :
( 1 ... ( N  +  1 ) ) -1-1-> A )
7168, 69, 70syl2anc 642 . . . . . . . 8  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( N  +  1 ) ) -1-1-> A )
72 simpr 447 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f  Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  s ) )
73 f1ofo 5495 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  s
) ) -1-1-onto-> s  ->  f : ( 1 ... ( # `
 s ) )
-onto-> s )
74 forn 5470 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  s
) ) -onto-> s  ->  ran  f  =  s
)
7556, 73, 743syl 18 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  ran  f  =  s )
76 isoeq5 5836 . . . . . . . . . . 11  |-  ( ran  f  =  s  -> 
( f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  ran  f )  <->  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) ) )
7775, 76syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( # `
 s ) ) ,  ran  f )  <-> 
f  Isom  <  ,  <  ( ( 1 ... ( # `
 s ) ) ,  s ) ) )
7872, 77mpbird 223 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f  Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  ran  f ) )
79 isoeq4 5835 . . . . . . . . . 10  |-  ( ( 1 ... ( # `  s ) )  =  ( 1 ... ( N  +  1 ) )  ->  ( f  Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  ran  f )  <-> 
f  Isom  <  ,  <  ( ( 1 ... ( N  +  1 ) ) ,  ran  f
) ) )
8063, 79syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( # `
 s ) ) ,  ran  f )  <-> 
f  Isom  <  ,  <  ( ( 1 ... ( N  +  1 ) ) ,  ran  f
) ) )
8178, 80mpbid 201 . . . . . . . 8  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f  Isom  <  ,  <  (
( 1 ... ( N  +  1 ) ) ,  ran  f
) )
8271, 81jca 518 . . . . . . 7  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
f : ( 1 ... ( N  + 
1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  +  1 ) ) ,  ran  f ) ) )
8382ex 423 . . . . . 6  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( # `
 s ) ) ,  s )  -> 
( f : ( 1 ... ( N  +  1 ) )
-1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  + 
1 ) ) ,  ran  f ) ) ) )
8483eximdv 1612 . . . . 5  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s )  ->  E. f
( f : ( 1 ... ( N  +  1 ) )
-1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  + 
1 ) ) ,  ran  f ) ) ) )
8554, 84mpd 14 . . . 4  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  E. f
( f : ( 1 ... ( N  +  1 ) )
-1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  + 
1 ) ) ,  ran  f ) ) )
8685ex 423 . . 3  |-  ( ph  ->  ( ( ( 1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
)  ->  E. f
( f : ( 1 ... ( N  +  1 ) )
-1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  + 
1 ) ) ,  ran  f ) ) ) )
8786exlimdv 1626 . 2  |-  ( ph  ->  ( E. s ( ( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A )  ->  E. f ( f : ( 1 ... ( N  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  (
( 1 ... ( N  +  1 ) ) ,  ran  f
) ) ) )
8841, 87mpd 14 1  |-  ( ph  ->  E. f ( f : ( 1 ... ( N  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  +  1 ) ) ,  ran  f
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    Or wor 4329   ran crn 4706   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272  (class class class)co 5874    ~~ cen 6876    ~<_ cdom 6877   Fincfn 6879   RRcr 8752   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   NN0cn0 9981   ...cfz 10798   #chash 11353
This theorem is referenced by:  erdsze2  23751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354
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