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Theorem cnre2csqlem 28225
Description: Lemma for cnre2csqima 28226 (Contributed by Thierry Arnoux, 27-Sep-2017.)
Hypotheses
Ref Expression
cnre2csqlem.1  |-  ( G  |`  ( RR  X.  RR ) )  =  ( H  o.  F )
cnre2csqlem.2  |-  F  Fn  ( RR  X.  RR )
cnre2csqlem.3  |-  G  Fn  _V
cnre2csqlem.4  |-  ( x  e.  ( RR  X.  RR )  ->  ( G `
 x )  e.  RR )
cnre2csqlem.5  |-  ( ( x  e.  ran  F  /\  y  e.  ran  F )  ->  ( H `  ( x  -  y
) )  =  ( ( H `  x
)  -  ( H `
 y ) ) )
Assertion
Ref Expression
cnre2csqlem  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( `' ( G  |`  ( RR 
X.  RR ) )
" ( ( ( G `  X )  -  D ) (,) ( ( G `  X )  +  D
) ) )  -> 
( abs `  ( H `  ( ( F `  Y )  -  ( F `  X ) ) ) )  <  D ) )
Distinct variable groups:    x, y, F    x, G    x, H, y    x, X, y    x, Y, y
Allowed substitution hints:    D( x, y)    G( y)

Proof of Theorem cnre2csqlem
StepHypRef Expression
1 cnre2csqlem.3 . . . . . . 7  |-  G  Fn  _V
2 ssv 3461 . . . . . . 7  |-  ( RR 
X.  RR )  C_  _V
3 fnssres 5631 . . . . . . 7  |-  ( ( G  Fn  _V  /\  ( RR  X.  RR )  C_  _V )  -> 
( G  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR ) )
41, 2, 3mp2an 670 . . . . . 6  |-  ( G  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR )
5 elpreima 5941 . . . . . 6  |-  ( ( G  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR )  ->  ( Y  e.  ( `' ( G  |`  ( RR  X.  RR ) ) " (
( ( G `  X )  -  D
) (,) ( ( G `  X )  +  D ) ) )  <->  ( Y  e.  ( RR  X.  RR )  /\  ( ( G  |`  ( RR  X.  RR ) ) `  Y
)  e.  ( ( ( G `  X
)  -  D ) (,) ( ( G `
 X )  +  D ) ) ) ) )
64, 5mp1i 13 . . . . 5  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( `' ( G  |`  ( RR 
X.  RR ) )
" ( ( ( G `  X )  -  D ) (,) ( ( G `  X )  +  D
) ) )  <->  ( Y  e.  ( RR  X.  RR )  /\  ( ( G  |`  ( RR  X.  RR ) ) `  Y
)  e.  ( ( ( G `  X
)  -  D ) (,) ( ( G `
 X )  +  D ) ) ) ) )
76simplbda 622 . . . 4  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  /\  Y  e.  ( `' ( G  |`  ( RR  X.  RR ) ) " (
( ( G `  X )  -  D
) (,) ( ( G `  X )  +  D ) ) ) )  ->  (
( G  |`  ( RR  X.  RR ) ) `
 Y )  e.  ( ( ( G `
 X )  -  D ) (,) (
( G `  X
)  +  D ) ) )
87ex 432 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( `' ( G  |`  ( RR 
X.  RR ) )
" ( ( ( G `  X )  -  D ) (,) ( ( G `  X )  +  D
) ) )  -> 
( ( G  |`  ( RR  X.  RR ) ) `  Y
)  e.  ( ( ( G `  X
)  -  D ) (,) ( ( G `
 X )  +  D ) ) ) )
9 simp2 998 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  Y  e.  ( RR  X.  RR ) )
10 fvres 5819 . . . . . 6  |-  ( Y  e.  ( RR  X.  RR )  ->  ( ( G  |`  ( RR  X.  RR ) ) `  Y )  =  ( G `  Y ) )
119, 10syl 17 . . . . 5  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G  |`  ( RR  X.  RR ) ) `
 Y )  =  ( G `  Y
) )
1211eleq1d 2471 . . . 4  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( ( G  |`  ( RR  X.  RR ) ) `  Y
)  e.  ( ( ( G `  X
)  -  D ) (,) ( ( G `
 X )  +  D ) )  <->  ( G `  Y )  e.  ( ( ( G `  X )  -  D
) (,) ( ( G `  X )  +  D ) ) ) )
13 simp1 997 . . . . . . . . . . . 12  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  X  e.  ( RR  X.  RR ) )
14 fveq2 5805 . . . . . . . . . . . . . 14  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1514eleq1d 2471 . . . . . . . . . . . . 13  |-  ( x  =  X  ->  (
( G `  x
)  e.  RR  <->  ( G `  X )  e.  RR ) )
16 cnre2csqlem.4 . . . . . . . . . . . . 13  |-  ( x  e.  ( RR  X.  RR )  ->  ( G `
 x )  e.  RR )
1715, 16vtoclga 3122 . . . . . . . . . . . 12  |-  ( X  e.  ( RR  X.  RR )  ->  ( G `
 X )  e.  RR )
1813, 17syl 17 . . . . . . . . . . 11  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( G `  X )  e.  RR )
19 simp3 999 . . . . . . . . . . . 12  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  D  e.  RR+ )
2019rpred 11222 . . . . . . . . . . 11  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  D  e.  RR )
2118, 20resubcld 9948 . . . . . . . . . 10  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  X
)  -  D )  e.  RR )
2221rexrd 9593 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  X
)  -  D )  e.  RR* )
2318, 20readdcld 9573 . . . . . . . . . 10  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  X
)  +  D )  e.  RR )
2423rexrd 9593 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  X
)  +  D )  e.  RR* )
25 elioo2 11541 . . . . . . . . 9  |-  ( ( ( ( G `  X )  -  D
)  e.  RR*  /\  (
( G `  X
)  +  D )  e.  RR* )  ->  (
( G `  Y
)  e.  ( ( ( G `  X
)  -  D ) (,) ( ( G `
 X )  +  D ) )  <->  ( ( G `  Y )  e.  RR  /\  ( ( G `  X )  -  D )  < 
( G `  Y
)  /\  ( G `  Y )  <  (
( G `  X
)  +  D ) ) ) )
2622, 24, 25syl2anc 659 . . . . . . . 8  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  Y
)  e.  ( ( ( G `  X
)  -  D ) (,) ( ( G `
 X )  +  D ) )  <->  ( ( G `  Y )  e.  RR  /\  ( ( G `  X )  -  D )  < 
( G `  Y
)  /\  ( G `  Y )  <  (
( G `  X
)  +  D ) ) ) )
2726biimpa 482 . . . . . . 7  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  /\  ( G `  Y )  e.  ( ( ( G `  X )  -  D
) (,) ( ( G `  X )  +  D ) ) )  ->  ( ( G `  Y )  e.  RR  /\  ( ( G `  X )  -  D )  < 
( G `  Y
)  /\  ( G `  Y )  <  (
( G `  X
)  +  D ) ) )
2827simp2d 1010 . . . . . 6  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  /\  ( G `  Y )  e.  ( ( ( G `  X )  -  D
) (,) ( ( G `  X )  +  D ) ) )  ->  ( ( G `  X )  -  D )  <  ( G `  Y )
)
2927simp3d 1011 . . . . . 6  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  /\  ( G `  Y )  e.  ( ( ( G `  X )  -  D
) (,) ( ( G `  X )  +  D ) ) )  ->  ( G `  Y )  <  (
( G `  X
)  +  D ) )
3028, 29jca 530 . . . . 5  |-  ( ( ( X  e.  ( RR  X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  /\  ( G `  Y )  e.  ( ( ( G `  X )  -  D
) (,) ( ( G `  X )  +  D ) ) )  ->  ( (
( G `  X
)  -  D )  <  ( G `  Y )  /\  ( G `  Y )  <  ( ( G `  X )  +  D
) ) )
3130ex 432 . . . 4  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  Y
)  e.  ( ( ( G `  X
)  -  D ) (,) ( ( G `
 X )  +  D ) )  -> 
( ( ( G `
 X )  -  D )  <  ( G `  Y )  /\  ( G `  Y
)  <  ( ( G `  X )  +  D ) ) ) )
3212, 31sylbid 215 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( ( G  |`  ( RR  X.  RR ) ) `  Y
)  e.  ( ( ( G `  X
)  -  D ) (,) ( ( G `
 X )  +  D ) )  -> 
( ( ( G `
 X )  -  D )  <  ( G `  Y )  /\  ( G `  Y
)  <  ( ( G `  X )  +  D ) ) ) )
33 fveq2 5805 . . . . . . 7  |-  ( x  =  Y  ->  ( G `  x )  =  ( G `  Y ) )
3433eleq1d 2471 . . . . . 6  |-  ( x  =  Y  ->  (
( G `  x
)  e.  RR  <->  ( G `  Y )  e.  RR ) )
3534, 16vtoclga 3122 . . . . 5  |-  ( Y  e.  ( RR  X.  RR )  ->  ( G `
 Y )  e.  RR )
369, 35syl 17 . . . 4  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( G `  Y )  e.  RR )
37 absdiflt 13206 . . . . 5  |-  ( ( ( G `  Y
)  e.  RR  /\  ( G `  X )  e.  RR  /\  D  e.  RR )  ->  (
( abs `  (
( G `  Y
)  -  ( G `
 X ) ) )  <  D  <->  ( (
( G `  X
)  -  D )  <  ( G `  Y )  /\  ( G `  Y )  <  ( ( G `  X )  +  D
) ) ) )
3837biimprd 223 . . . 4  |-  ( ( ( G `  Y
)  e.  RR  /\  ( G `  X )  e.  RR  /\  D  e.  RR )  ->  (
( ( ( G `
 X )  -  D )  <  ( G `  Y )  /\  ( G `  Y
)  <  ( ( G `  X )  +  D ) )  -> 
( abs `  (
( G `  Y
)  -  ( G `
 X ) ) )  <  D ) )
3936, 18, 20, 38syl3anc 1230 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( ( ( G `
 X )  -  D )  <  ( G `  Y )  /\  ( G `  Y
)  <  ( ( G `  X )  +  D ) )  -> 
( abs `  (
( G `  Y
)  -  ( G `
 X ) ) )  <  D ) )
408, 32, 393syld 54 . 2  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( `' ( G  |`  ( RR 
X.  RR ) )
" ( ( ( G `  X )  -  D ) (,) ( ( G `  X )  +  D
) ) )  -> 
( abs `  (
( G `  Y
)  -  ( G `
 X ) ) )  <  D ) )
41 cnre2csqlem.2 . . . . . . 7  |-  F  Fn  ( RR  X.  RR )
42 fnfvelrn 5962 . . . . . . 7  |-  ( ( F  Fn  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR ) )  -> 
( F `  Y
)  e.  ran  F
)
4341, 9, 42sylancr 661 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( F `  Y )  e.  ran  F )
44 fnfvelrn 5962 . . . . . . 7  |-  ( ( F  Fn  ( RR 
X.  RR )  /\  X  e.  ( RR  X.  RR ) )  -> 
( F `  X
)  e.  ran  F
)
4541, 13, 44sylancr 661 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( F `  X )  e.  ran  F )
46 oveq1 6241 . . . . . . . . 9  |-  ( x  =  ( F `  Y )  ->  (
x  -  y )  =  ( ( F `
 Y )  -  y ) )
4746fveq2d 5809 . . . . . . . 8  |-  ( x  =  ( F `  Y )  ->  ( H `  ( x  -  y ) )  =  ( H `  ( ( F `  Y )  -  y
) ) )
48 fveq2 5805 . . . . . . . . 9  |-  ( x  =  ( F `  Y )  ->  ( H `  x )  =  ( H `  ( F `  Y ) ) )
4948oveq1d 6249 . . . . . . . 8  |-  ( x  =  ( F `  Y )  ->  (
( H `  x
)  -  ( H `
 y ) )  =  ( ( H `
 ( F `  Y ) )  -  ( H `  y ) ) )
5047, 49eqeq12d 2424 . . . . . . 7  |-  ( x  =  ( F `  Y )  ->  (
( H `  (
x  -  y ) )  =  ( ( H `  x )  -  ( H `  y ) )  <->  ( H `  ( ( F `  Y )  -  y
) )  =  ( ( H `  ( F `  Y )
)  -  ( H `
 y ) ) ) )
51 oveq2 6242 . . . . . . . . 9  |-  ( y  =  ( F `  X )  ->  (
( F `  Y
)  -  y )  =  ( ( F `
 Y )  -  ( F `  X ) ) )
5251fveq2d 5809 . . . . . . . 8  |-  ( y  =  ( F `  X )  ->  ( H `  ( ( F `  Y )  -  y ) )  =  ( H `  ( ( F `  Y )  -  ( F `  X )
) ) )
53 fveq2 5805 . . . . . . . . 9  |-  ( y  =  ( F `  X )  ->  ( H `  y )  =  ( H `  ( F `  X ) ) )
5453oveq2d 6250 . . . . . . . 8  |-  ( y  =  ( F `  X )  ->  (
( H `  ( F `  Y )
)  -  ( H `
 y ) )  =  ( ( H `
 ( F `  Y ) )  -  ( H `  ( F `
 X ) ) ) )
5552, 54eqeq12d 2424 . . . . . . 7  |-  ( y  =  ( F `  X )  ->  (
( H `  (
( F `  Y
)  -  y ) )  =  ( ( H `  ( F `
 Y ) )  -  ( H `  y ) )  <->  ( H `  ( ( F `  Y )  -  ( F `  X )
) )  =  ( ( H `  ( F `  Y )
)  -  ( H `
 ( F `  X ) ) ) ) )
56 cnre2csqlem.5 . . . . . . 7  |-  ( ( x  e.  ran  F  /\  y  e.  ran  F )  ->  ( H `  ( x  -  y
) )  =  ( ( H `  x
)  -  ( H `
 y ) ) )
5750, 55, 56vtocl2ga 3124 . . . . . 6  |-  ( ( ( F `  Y
)  e.  ran  F  /\  ( F `  X
)  e.  ran  F
)  ->  ( H `  ( ( F `  Y )  -  ( F `  X )
) )  =  ( ( H `  ( F `  Y )
)  -  ( H `
 ( F `  X ) ) ) )
5843, 45, 57syl2anc 659 . . . . 5  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( H `  ( ( F `  Y )  -  ( F `  X ) ) )  =  ( ( H `
 ( F `  Y ) )  -  ( H `  ( F `
 X ) ) ) )
59 cnre2csqlem.1 . . . . . . . 8  |-  ( G  |`  ( RR  X.  RR ) )  =  ( H  o.  F )
6059fveq1i 5806 . . . . . . 7  |-  ( ( G  |`  ( RR  X.  RR ) ) `  Y )  =  ( ( H  o.  F
) `  Y )
61 fvco2 5880 . . . . . . . 8  |-  ( ( F  Fn  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR ) )  -> 
( ( H  o.  F ) `  Y
)  =  ( H `
 ( F `  Y ) ) )
6241, 9, 61sylancr 661 . . . . . . 7  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( H  o.  F
) `  Y )  =  ( H `  ( F `  Y ) ) )
6360, 11, 623eqtr3a 2467 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( G `  Y )  =  ( H `  ( F `  Y ) ) )
6459fveq1i 5806 . . . . . . 7  |-  ( ( G  |`  ( RR  X.  RR ) ) `  X )  =  ( ( H  o.  F
) `  X )
65 fvres 5819 . . . . . . . 8  |-  ( X  e.  ( RR  X.  RR )  ->  ( ( G  |`  ( RR  X.  RR ) ) `  X )  =  ( G `  X ) )
6613, 65syl 17 . . . . . . 7  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G  |`  ( RR  X.  RR ) ) `
 X )  =  ( G `  X
) )
67 fvco2 5880 . . . . . . . 8  |-  ( ( F  Fn  ( RR 
X.  RR )  /\  X  e.  ( RR  X.  RR ) )  -> 
( ( H  o.  F ) `  X
)  =  ( H `
 ( F `  X ) ) )
6841, 13, 67sylancr 661 . . . . . . 7  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( H  o.  F
) `  X )  =  ( H `  ( F `  X ) ) )
6964, 66, 683eqtr3a 2467 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( G `  X )  =  ( H `  ( F `  X ) ) )
7063, 69oveq12d 6252 . . . . 5  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  Y
)  -  ( G `
 X ) )  =  ( ( H `
 ( F `  Y ) )  -  ( H `  ( F `
 X ) ) ) )
7158, 70eqtr4d 2446 . . . 4  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( H `  ( ( F `  Y )  -  ( F `  X ) ) )  =  ( ( G `
 Y )  -  ( G `  X ) ) )
7271fveq2d 5809 . . 3  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( abs `  ( H `  ( ( F `  Y )  -  ( F `  X )
) ) )  =  ( abs `  (
( G `  Y
)  -  ( G `
 X ) ) ) )
7372breq1d 4404 . 2  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( abs `  ( H `  ( ( F `  Y )  -  ( F `  X ) ) ) )  <  D  <->  ( abs `  ( ( G `  Y )  -  ( G `  X )
) )  <  D
) )
7440, 73sylibrd 234 1  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( `' ( G  |`  ( RR 
X.  RR ) )
" ( ( ( G `  X )  -  D ) (,) ( ( G `  X )  +  D
) ) )  -> 
( abs `  ( H `  ( ( F `  Y )  -  ( F `  X ) ) ) )  <  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   _Vcvv 3058    C_ wss 3413   class class class wbr 4394    X. cxp 4940   `'ccnv 4941   ran crn 4943    |` cres 4944   "cima 4945    o. ccom 4946    Fn wfn 5520   ` cfv 5525  (class class class)co 6234   RRcr 9441    + caddc 9445   RR*cxr 9577    < clt 9578    - cmin 9761   RR+crp 11183   (,)cioo 11500   abscabs 13123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-sup 7855  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-rp 11184  df-ioo 11504  df-seq 12062  df-exp 12121  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125
This theorem is referenced by:  cnre2csqima  28226
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