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Theorem tgqioo 12466
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 9531 . . . 4  |-  (,)  e.  _V
32imaex 4830 . . 3  |-  ( (,) " ( QQ  X.  QQ ) )  e.  _V
4 imassrn 4828 . . 3  |-  ( (,) " ( QQ  X.  QQ ) )  C_  ran  (,)
5 ioof 9595 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
6 ffn 5208 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
75, 6ax-mp 7 . . . . 5  |-  (,)  Fn  ( RR*  X.  RR* )
8 simpll 499 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  x  e.  RR* )
9 elioo1 9535 . . . . . . . . . . . 12  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  (
z  e.  ( x (,) y )  <->  ( z  e.  RR*  /\  x  < 
z  /\  z  <  y ) ) )
109biimpa 292 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  ( z  e.  RR*  /\  x  < 
z  /\  z  <  y ) )
1110simp1d 961 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  z  e.  RR* )
1210simp2d 962 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  x  <  z )
13 qbtwnxr 9876 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  z  e.  RR*  /\  x  < 
z )  ->  E. u  e.  QQ  ( x  < 
u  /\  u  <  z ) )
148, 11, 12, 13syl3anc 1184 . . . . . . . . 9  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  E. u  e.  QQ  ( x  < 
u  /\  u  <  z ) )
15 simplr 500 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  y  e.  RR* )
1610simp3d 963 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  z  <  y )
17 qbtwnxr 9876 . . . . . . . . . 10  |-  ( ( z  e.  RR*  /\  y  e.  RR*  /\  z  < 
y )  ->  E. v  e.  QQ  ( z  < 
v  /\  v  <  y ) )
1811, 15, 16, 17syl3anc 1184 . . . . . . . . 9  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  z  e.  (
x (,) y ) )  ->  E. v  e.  QQ  ( z  < 
v  /\  v  <  y ) )
19 reeanv 2558 . . . . . . . . . 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 5709 . . . . . . . . . . . . . 14  |-  ( u (,) v )  =  ( (,) `  <. u ,  v >. )
21 opelxpi 4509 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  QQ  /\  v  e.  QQ )  -> 
<. u ,  v >.  e.  ( QQ  X.  QQ ) )
22213ad2ant2 971 . . . . . . . . . . . . . . 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 5211 . . . . . . . . . . . . . . . . 17  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  Fun  (,) )
245, 23ax-mp 7 . . . . . . . . . . . . . . . 16  |-  Fun  (,)
25 qssre 9272 . . . . . . . . . . . . . . . . . . 19  |-  QQ  C_  RR
26 ressxr 7681 . . . . . . . . . . . . . . . . . . 19  |-  RR  C_  RR*
2725, 26sstri 3056 . . . . . . . . . . . . . . . . . 18  |-  QQ  C_  RR*
28 xpss12 4584 . . . . . . . . . . . . . . . . . 18  |-  ( ( QQ  C_  RR*  /\  QQ  C_ 
RR* )  ->  ( QQ  X.  QQ )  C_  ( RR*  X.  RR* )
)
2927, 27, 28mp2an 420 . . . . . . . . . . . . . . . . 17  |-  ( QQ 
X.  QQ )  C_  ( RR*  X.  RR* )
305fdmi 5216 . . . . . . . . . . . . . . . . 17  |-  dom  (,)  =  ( RR*  X.  RR* )
3129, 30sseqtr4i 3082 . . . . . . . . . . . . . . . 16  |-  ( QQ 
X.  QQ )  C_  dom  (,)
32 funfvima2 5582 . . . . . . . . . . . . . . . 16  |-  ( ( Fun  (,)  /\  ( QQ  X.  QQ )  C_  dom  (,) )  ->  ( <. u ,  v >.  e.  ( QQ  X.  QQ )  ->  ( (,) `  <. u ,  v >. )  e.  ( (,) " ( QQ  X.  QQ ) ) ) )
3324, 31, 32mp2an 420 . . . . . . . . . . . . . . 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, 34syl5eqel 2186 . . . . . . . . . . . . 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 970 . . . . . . . . . . . . . 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 1021 . . . . . . . . . . . . . 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 1022 . . . . . . . . . . . . . 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 975 . . . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . . . 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 976 . . . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . . . 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 9535 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  RR*  /\  v  e.  RR* )  ->  (
z  e.  ( u (,) v )  <->  ( z  e.  RR*  /\  u  < 
z  /\  z  <  v ) ) )
4440, 42, 43syl2anc 406 . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . 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 970 . . . . . . . . . . . . . . 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 1020 . . . . . . . . . . . . . . . 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 9426 . . . . . . . . . . . . . . 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 9540 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  x  <_  u )  ->  (
u (,) v ) 
C_  ( x (,) v ) )
5046, 48, 49syl2anc 406 . . . . . . . . . . . . . 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 970 . . . . . . . . . . . . . . 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 1023 . . . . . . . . . . . . . . . 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 9426 . . . . . . . . . . . . . . 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 9541 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR*  /\  v  <_  y )  ->  (
x (,) v ) 
C_  ( x (,) y ) )
5551, 53, 54syl2anc 406 . . . . . . . . . . . . . 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 3057 . . . . . . . . . . . . 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 2163 . . . . . . . . . . . . . . 15  |-  ( w  =  ( u (,) v )  ->  (
z  e.  w  <->  z  e.  ( u (,) v
) ) )
58 sseq1 3070 . . . . . . . . . . . . . . 15  |-  ( w  =  ( u (,) v )  ->  (
w  C_  ( x (,) y )  <->  ( u (,) v )  C_  (
x (,) y ) ) )
5957, 58anbi12d 460 . . . . . . . . . . . . . 14  |-  ( w  =  ( u (,) v )  ->  (
( z  e.  w  /\  w  C_  ( x (,) y ) )  <-> 
( z  e.  ( u (,) v )  /\  ( u (,) v )  C_  (
x (,) y ) ) ) )
6059rspcev 2744 . . . . . . . . . . . . 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 1182 . . . . . . . . . . . 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 1148 . . . . . . . . . . 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 2515 . . . . . . . . . 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 427 . . . . . . . 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 2464 . . . . . . 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 12444 . . . . . . . 8  |-  ( (,) " ( QQ  X.  QQ ) )  e.  TopBases
68 eltg2b 12005 . . . . . . . 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 7 . . . . . . 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 2445 . . . . 5  |-  A. x  e.  RR*  A. y  e. 
RR*  ( x (,) y )  e.  (
topGen `  ( (,) " ( QQ  X.  QQ ) ) )
72 ffnov 5807 . . . . 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 894 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )
74 frn 5217 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  ->  ran  (,)  C_  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) ) )
7573, 74ax-mp 7 . . 3  |-  ran  (,)  C_  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )
76 2basgeng 12033 . . 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 1283 . 2  |-  ( topGen `  ( (,) " ( QQ  X.  QQ ) ) )  =  ( topGen ` 
ran  (,) )
781, 77eqtr2i 2121 1  |-  ( topGen ` 
ran  (,) )  =  Q
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 930    = wceq 1299    e. wcel 1448   A.wral 2375   E.wrex 2376   _Vcvv 2641    C_ wss 3021   ~Pcpw 3457   <.cop 3477   class class class wbr 3875    X. cxp 4475   dom cdm 4477   ran crn 4478   "cima 4480   Fun wfun 5053    Fn wfn 5054   -->wf 5055   ` cfv 5059  (class class class)co 5706   RRcr 7499   RR*cxr 7671    < clt 7672    <_ cle 7673   QQcq 9261   (,)cioo 9512   topGenctg 11917   TopBasesctb 11991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-sup 6786  df-inf 6787  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-xneg 9400  df-ioo 9516  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-topgen 11923  df-bases 11992
This theorem is referenced by: (None)
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