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Theorem tgqioo 12753
Description: The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)
Hypothesis
Ref Expression
tgqioo.1  |-  Q  =  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )
Assertion
Ref Expression
tgqioo  |-  ( topGen ` 
ran  (,) )  =  Q

Proof of Theorem tgqioo
Dummy variables  v  u  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgqioo.1 . 2  |-  Q  =  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )
2 iooex 9719 . . . 4  |-  (,)  e.  _V
32imaex 4901 . . 3  |-  ( (,) " ( QQ  X.  QQ ) )  e.  _V
4 imassrn 4899 . . 3  |-  ( (,) " ( QQ  X.  QQ ) )  C_  ran  (,)
5 ioof 9783 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
6 ffn 5279 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
75, 6ax-mp 5 . . . . 5  |-  (,)  Fn  ( RR*  X.  RR* )
8 simpll 519 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  x  e.  RR* )
9 elioo1 9723 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
z  e.  ( x (,) y )  <->  ( z  e.  RR*  /\  x  < 
z  /\  z  <  y ) ) )
109biimpa 294 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  ( z  e.  RR*  /\  x  < 
z  /\  z  <  y ) )
1110simp1d 994 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  z  e.  RR* )
1210simp2d 995 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  x  <  z )
13 qbtwnxr 10065 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  z  e.  RR*  /\  x  < 
z )  ->  E. u  e.  QQ  ( x  < 
u  /\  u  <  z ) )
148, 11, 12, 13syl3anc 1217 . . . . . . . . 9  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  E. u  e.  QQ  ( x  < 
u  /\  u  <  z ) )
15 simplr 520 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  y  e.  RR* )
1610simp3d 996 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  z  <  y )
17 qbtwnxr 10065 . . . . . . . . . 10  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  z  < 
y )  ->  E. v  e.  QQ  ( z  < 
v  /\  v  <  y ) )
1811, 15, 16, 17syl3anc 1217 . . . . . . . . 9  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  E. v  e.  QQ  ( z  < 
v  /\  v  <  y ) )
19 reeanv 2603 . . . . . . . . . 10  |-  ( E. u  e.  QQ  E. v  e.  QQ  (
( x  <  u  /\  u  <  z )  /\  ( z  < 
v  /\  v  <  y ) )  <->  ( E. u  e.  QQ  (
x  <  u  /\  u  <  z )  /\  E. v  e.  QQ  (
z  <  v  /\  v  <  y ) ) )
20 df-ov 5784 . . . . . . . . . . . . . 14  |-  ( u (,) v )  =  ( (,) `  <. u ,  v >. )
21 opelxpi 4578 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  QQ  /\  v  e.  QQ )  -> 
<. u ,  v >.  e.  ( QQ  X.  QQ ) )
22213ad2ant2 1004 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  <. u ,  v
>.  e.  ( QQ  X.  QQ ) )
23 ffun 5282 . . . . . . . . . . . . . . . . 17  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
245, 23ax-mp 5 . . . . . . . . . . . . . . . 16  |-  Fun  (,)
25 qssre 9448 . . . . . . . . . . . . . . . . . . 19  |-  QQ  C_  RR
26 ressxr 7832 . . . . . . . . . . . . . . . . . . 19  |-  RR  C_  RR*
2725, 26sstri 3110 . . . . . . . . . . . . . . . . . 18  |-  QQ  C_  RR*
28 xpss12 4653 . . . . . . . . . . . . . . . . . 18  |-  ( ( QQ  C_  RR*  /\  QQ  C_ 
RR* )  ->  ( QQ  X.  QQ )  C_  ( RR*  X.  RR* )
)
2927, 27, 28mp2an 423 . . . . . . . . . . . . . . . . 17  |-  ( QQ 
X.  QQ )  C_  ( RR*  X.  RR* )
305fdmi 5287 . . . . . . . . . . . . . . . . 17  |-  dom  (,)  =  ( RR*  X.  RR* )
3129, 30sseqtrri 3136 . . . . . . . . . . . . . . . 16  |-  ( QQ 
X.  QQ )  C_  dom  (,)
32 funfvima2 5657 . . . . . . . . . . . . . . . 16  |-  ( ( Fun  (,)  /\  ( QQ  X.  QQ )  C_  dom  (,) )  ->  ( <. u ,  v >.  e.  ( QQ  X.  QQ )  ->  ( (,) `  <. u ,  v >. )  e.  ( (,) " ( QQ  X.  QQ ) ) ) )
3324, 31, 32mp2an 423 . . . . . . . . . . . . . . 15  |-  ( <.
u ,  v >.  e.  ( QQ  X.  QQ )  ->  ( (,) `  <. u ,  v >. )  e.  ( (,) " ( QQ  X.  QQ ) ) )
3422, 33syl 14 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( (,) `  <. u ,  v >. )  e.  ( (,) " ( QQ  X.  QQ ) ) )
3520, 34eqeltrid 2227 . . . . . . . . . . . . 13  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( u (,) v )  e.  ( (,) " ( QQ 
X.  QQ ) ) )
36113ad2ant1 1003 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  z  e.  RR* )
37 simp3lr 1054 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  u  <  z
)
38 simp3rl 1055 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  z  <  v
)
39 simp2l 1008 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  u  e.  QQ )
4027, 39sseldi 3099 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  u  e.  RR* )
41 simp2r 1009 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  v  e.  QQ )
4227, 41sseldi 3099 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  v  e.  RR* )
43 elioo1 9723 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  RR*  /\  v  e.  RR* )  ->  (
z  e.  ( u (,) v )  <->  ( z  e.  RR*  /\  u  < 
z  /\  z  <  v ) ) )
4440, 42, 43syl2anc 409 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( z  e.  ( u (,) v
)  <->  ( z  e. 
RR*  /\  u  <  z  /\  z  <  v
) ) )
4536, 37, 38, 44mpbir3and 1165 . . . . . . . . . . . . 13  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  z  e.  ( u (,) v ) )
4683ad2ant1 1003 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  x  e.  RR* )
47 simp3ll 1053 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  x  <  u
)
4846, 40, 47xrltled 9614 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  x  <_  u
)
49 iooss1 9728 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  x  <_  u )  ->  (
u (,) v ) 
C_  ( x (,) v ) )
5046, 48, 49syl2anc 409 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( u (,) v )  C_  (
x (,) v ) )
51153ad2ant1 1003 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  y  e.  RR* )
52 simp3rr 1056 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  v  <  y
)
5342, 51, 52xrltled 9614 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  v  <_  y
)
54 iooss2 9729 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  v  <_  y )  ->  (
x (,) v ) 
C_  ( x (,) y ) )
5551, 53, 54syl2anc 409 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( x (,) v )  C_  (
x (,) y ) )
5650, 55sstrd 3111 . . . . . . . . . . . . 13  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  ( u (,) v )  C_  (
x (,) y ) )
57 eleq2 2204 . . . . . . . . . . . . . . 15  |-  ( w  =  ( u (,) v )  ->  (
z  e.  w  <->  z  e.  ( u (,) v
) ) )
58 sseq1 3124 . . . . . . . . . . . . . . 15  |-  ( w  =  ( u (,) v )  ->  (
w  C_  ( x (,) y )  <->  ( u (,) v )  C_  (
x (,) y ) ) )
5957, 58anbi12d 465 . . . . . . . . . . . . . 14  |-  ( w  =  ( u (,) v )  ->  (
( z  e.  w  /\  w  C_  ( x (,) y ) )  <-> 
( z  e.  ( u (,) v )  /\  ( u (,) v )  C_  (
x (,) y ) ) ) )
6059rspcev 2792 . . . . . . . . . . . . 13  |-  ( ( ( u (,) v
)  e.  ( (,) " ( QQ  X.  QQ ) )  /\  (
z  e.  ( u (,) v )  /\  ( u (,) v
)  C_  ( x (,) y ) ) )  ->  E. w  e.  ( (,) " ( QQ 
X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) )
6135, 45, 56, 60syl12anc 1215 . . . . . . . . . . . 12  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  z  e.  ( x (,) y
) )  /\  (
u  e.  QQ  /\  v  e.  QQ )  /\  ( ( x  < 
u  /\  u  <  z )  /\  ( z  <  v  /\  v  <  y ) ) )  ->  E. w  e.  ( (,) " ( QQ 
X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) )
62613exp 1181 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  ( (
u  e.  QQ  /\  v  e.  QQ )  ->  ( ( ( x  <  u  /\  u  <  z )  /\  (
z  <  v  /\  v  <  y ) )  ->  E. w  e.  ( (,) " ( QQ 
X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) ) ) )
6362rexlimdvv 2559 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  ( E. u  e.  QQ  E. v  e.  QQ  ( ( x  <  u  /\  u  <  z )  /\  (
z  <  v  /\  v  <  y ) )  ->  E. w  e.  ( (,) " ( QQ 
X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) ) )
6419, 63syl5bir 152 . . . . . . . . 9  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  ( ( E. u  e.  QQ  ( x  <  u  /\  u  <  z )  /\  E. v  e.  QQ  (
z  <  v  /\  v  <  y ) )  ->  E. w  e.  ( (,) " ( QQ 
X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) ) )
6514, 18, 64mp2and 430 . . . . . . . 8  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  E. w  e.  ( (,) " ( QQ  X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) )
6665ralrimiva 2508 . . . . . . 7  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  A. z  e.  ( x (,) y
) E. w  e.  ( (,) " ( QQ  X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) )
67 qtopbas 12728 . . . . . . . 8  |-  ( (,) " ( QQ  X.  QQ ) )  e.  TopBases
68 eltg2b 12260 . . . . . . . 8  |-  ( ( (,) " ( QQ 
X.  QQ ) )  e.  TopBases  ->  ( ( x (,) y )  e.  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  <->  A. z  e.  ( x (,) y
) E. w  e.  ( (,) " ( QQ  X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) ) )
6967, 68ax-mp 5 . . . . . . 7  |-  ( ( x (,) y )  e.  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  <->  A. z  e.  ( x (,) y
) E. w  e.  ( (,) " ( QQ  X.  QQ ) ) ( z  e.  w  /\  w  C_  ( x (,) y ) ) )
7066, 69sylibr 133 . . . . . 6  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
x (,) y )  e.  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) ) )
7170rgen2a 2489 . . . . 5  |-  A. x  e.  RR*  A. y  e. 
RR*  ( x (,) y )  e.  (
topGen `  ( (,) " ( QQ  X.  QQ ) ) )
72 ffnov 5882 . . . . 5  |-  ( (,)
: ( RR*  X.  RR* )
--> ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  <->  ( (,)  Fn  ( RR*  X.  RR* )  /\  A. x  e.  RR*  A. y  e.  RR*  (
x (,) y )  e.  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) ) ) )
737, 71, 72mpbir2an 927 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )
74 frn 5288 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  ->  ran  (,)  C_  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) ) )
7573, 74ax-mp 5 . . 3  |-  ran  (,)  C_  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )
76 2basgeng 12288 . . 3  |-  ( ( ( (,) " ( QQ  X.  QQ ) )  e.  _V  /\  ( (,) " ( QQ  X.  QQ ) )  C_  ran  (,) 
/\  ran  (,)  C_  ( topGen `
 ( (,) " ( QQ  X.  QQ ) ) ) )  ->  ( topGen `
 ( (,) " ( QQ  X.  QQ ) ) )  =  ( topGen ` 
ran  (,) ) )
773, 4, 75, 76mp3an 1316 . 2  |-  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  =  ( topGen ` 
ran  (,) )
781, 77eqtr2i 2162 1  |-  ( topGen ` 
ran  (,) )  =  Q
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   _Vcvv 2689    C_ wss 3075   ~Pcpw 3514   <.cop 3534   class class class wbr 3936    X. cxp 4544   dom cdm 4546   ran crn 4547   "cima 4549   Fun wfun 5124    Fn wfn 5125   -->wf 5126   ` cfv 5130  (class class class)co 5781   RRcr 7642   RR*cxr 7822    < clt 7823    <_ cle 7824   QQcq 9437   (,)cioo 9700   topGenctg 12172   TopBasesctb 12246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762  ax-caucvg 7763
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-isom 5139  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-sup 6878  df-inf 6879  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078  df-uz 9350  df-q 9438  df-rp 9470  df-xneg 9588  df-ioo 9704  df-seqfrec 10249  df-exp 10323  df-cj 10645  df-re 10646  df-im 10647  df-rsqrt 10801  df-abs 10802  df-topgen 12178  df-bases 12247
This theorem is referenced by: (None)
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