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Theorem erdsze2lem1 23734
Description: Lemma for erdsze2 23736. (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 10005 . . . . . . . . . . 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 10005 . . . . . . . . . . 11  |-  ( S  e.  NN  ->  ( S  -  1 )  e.  NN0 )
75, 6syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( S  -  1 )  e.  NN0 )
84, 7nn0mulcld 10023 . . . . . . . . 9  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  e.  NN0 )
91, 8syl5eqel 2367 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
10 peano2nn0 10004 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
11 hashfz1 11345 . . . . . . . 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 11350 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
17 nn0ltp1le 10074 . . . . . . . 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 4043 . . . . 5  |-  ( (
ph  /\  A  e.  Fin )  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  <_  ( # `
 A ) )
21 fzfid 11035 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( 1 ... ( N  + 
1 ) )  e. 
Fin )
22 simpr 447 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  A  e. 
Fin )
23 hashdom 11361 . . . . . 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 11035 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  (
1 ... ( N  + 
1 ) )  e. 
Fin )
28 isinffi 7625 . . . . . 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 8828 . . . . . . . 8  |-  RR  e.  _V
32 ssexg 4160 . . . . . . . 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 6872 . . . . . 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 6876 . . . 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 3189 . . . . . . 7  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  s  C_  RR )
45 ltso 8903 . . . . . . 7  |-  <  Or  RR
46 soss 4332 . . . . . . 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 11035 . . . . . . 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 7079 . . . . . . . 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 11400 . . . . . 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 5822 . . . . . . . . . . . 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 11346 . . . . . . . . . . . . . . . . 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 2317 . . . . . . . . . . . . . 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 5874 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
1 ... ( # `  s
) )  =  ( 1 ... ( N  +  1 ) ) )
64 f1oeq2 5464 . . . . . . . . . . . 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 5471 . . . . . . . . . 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 5442 . . . . . . . . 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 5479 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  s
) ) -1-1-onto-> s  ->  f : ( 1 ... ( # `
 s ) )
-onto-> s )
74 forn 5454 . . . . . . . . . . . 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 5820 . . . . . . . . . . 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 5819 . . . . . . . . . 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 1608 . . . . 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 1664 . 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    Or wor 4313   ran crn 4690   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256  (class class class)co 5858    ~~ cen 6860    ~<_ cdom 6861   Fincfn 6863   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   NN0cn0 9965   ...cfz 10782   #chash 11337
This theorem is referenced by:  erdsze2  23736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-hash 11338
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