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Theorem cnre2csqlem 28724
Description: Lemma for cnre2csqima 28725 (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 3484 . . . . . . 7  |-  ( RR 
X.  RR )  C_  _V
3 fnssres 5707 . . . . . . 7  |-  ( ( G  Fn  _V  /\  ( RR  X.  RR )  C_  _V )  -> 
( G  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR ) )
41, 2, 3mp2an 676 . . . . . 6  |-  ( G  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR )
5 elpreima 6017 . . . . . 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 628 . . . 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 435 . . 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 1006 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  Y  e.  ( RR  X.  RR ) )
10 fvres 5895 . . . . . 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 2491 . . . 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 1005 . . . . . . . . . . . 12  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  X  e.  ( RR  X.  RR ) )
14 fveq2 5881 . . . . . . . . . . . . . 14  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1514eleq1d 2491 . . . . . . . . . . . . 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 3145 . . . . . . . . . . . 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 1007 . . . . . . . . . . . 12  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  D  e.  RR+ )
2019rpred 11348 . . . . . . . . . . 11  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  D  e.  RR )
2118, 20resubcld 10054 . . . . . . . . . 10  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  X
)  -  D )  e.  RR )
2221rexrd 9697 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  X
)  -  D )  e.  RR* )
2318, 20readdcld 9677 . . . . . . . . . 10  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  X
)  +  D )  e.  RR )
2423rexrd 9697 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  X
)  +  D )  e.  RR* )
25 elioo2 11684 . . . . . . . . 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 665 . . . . . . . 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 486 . . . . . . 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 1018 . . . . . 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 1019 . . . . . 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 534 . . . . 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 435 . . . 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 218 . . 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 5881 . . . . . . 7  |-  ( x  =  Y  ->  ( G `  x )  =  ( G `  Y ) )
3433eleq1d 2491 . . . . . 6  |-  ( x  =  Y  ->  (
( G `  x
)  e.  RR  <->  ( G `  Y )  e.  RR ) )
3534, 16vtoclga 3145 . . . . 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 13380 . . . . 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 226 . . . 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 1264 . . 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 57 . 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 6034 . . . . . . 7  |-  ( ( F  Fn  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR ) )  -> 
( F `  Y
)  e.  ran  F
)
4341, 9, 42sylancr 667 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( F `  Y )  e.  ran  F )
44 fnfvelrn 6034 . . . . . . 7  |-  ( ( F  Fn  ( RR 
X.  RR )  /\  X  e.  ( RR  X.  RR ) )  -> 
( F `  X
)  e.  ran  F
)
4541, 13, 44sylancr 667 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( F `  X )  e.  ran  F )
46 oveq1 6312 . . . . . . . . 9  |-  ( x  =  ( F `  Y )  ->  (
x  -  y )  =  ( ( F `
 Y )  -  y ) )
4746fveq2d 5885 . . . . . . . 8  |-  ( x  =  ( F `  Y )  ->  ( H `  ( x  -  y ) )  =  ( H `  ( ( F `  Y )  -  y
) ) )
48 fveq2 5881 . . . . . . . . 9  |-  ( x  =  ( F `  Y )  ->  ( H `  x )  =  ( H `  ( F `  Y ) ) )
4948oveq1d 6320 . . . . . . . 8  |-  ( x  =  ( F `  Y )  ->  (
( H `  x
)  -  ( H `
 y ) )  =  ( ( H `
 ( F `  Y ) )  -  ( H `  y ) ) )
5047, 49eqeq12d 2444 . . . . . . 7  |-  ( x  =  ( F `  Y )  ->  (
( H `  (
x  -  y ) )  =  ( ( H `  x )  -  ( H `  y ) )  <->  ( H `  ( ( F `  Y )  -  y
) )  =  ( ( H `  ( F `  Y )
)  -  ( H `
 y ) ) ) )
51 oveq2 6313 . . . . . . . . 9  |-  ( y  =  ( F `  X )  ->  (
( F `  Y
)  -  y )  =  ( ( F `
 Y )  -  ( F `  X ) ) )
5251fveq2d 5885 . . . . . . . 8  |-  ( y  =  ( F `  X )  ->  ( H `  ( ( F `  Y )  -  y ) )  =  ( H `  ( ( F `  Y )  -  ( F `  X )
) ) )
53 fveq2 5881 . . . . . . . . 9  |-  ( y  =  ( F `  X )  ->  ( H `  y )  =  ( H `  ( F `  X ) ) )
5453oveq2d 6321 . . . . . . . 8  |-  ( y  =  ( F `  X )  ->  (
( H `  ( F `  Y )
)  -  ( H `
 y ) )  =  ( ( H `
 ( F `  Y ) )  -  ( H `  ( F `
 X ) ) ) )
5552, 54eqeq12d 2444 . . . . . . 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 3147 . . . . . 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 665 . . . . 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 5882 . . . . . . 7  |-  ( ( G  |`  ( RR  X.  RR ) ) `  Y )  =  ( ( H  o.  F
) `  Y )
61 fvco2 5956 . . . . . . . 8  |-  ( ( F  Fn  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR ) )  -> 
( ( H  o.  F ) `  Y
)  =  ( H `
 ( F `  Y ) ) )
6241, 9, 61sylancr 667 . . . . . . 7  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( H  o.  F
) `  Y )  =  ( H `  ( F `  Y ) ) )
6360, 11, 623eqtr3a 2487 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( G `  Y )  =  ( H `  ( F `  Y ) ) )
6459fveq1i 5882 . . . . . . 7  |-  ( ( G  |`  ( RR  X.  RR ) ) `  X )  =  ( ( H  o.  F
) `  X )
65 fvres 5895 . . . . . . . 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 5956 . . . . . . . 8  |-  ( ( F  Fn  ( RR 
X.  RR )  /\  X  e.  ( RR  X.  RR ) )  -> 
( ( H  o.  F ) `  X
)  =  ( H `
 ( F `  X ) ) )
6841, 13, 67sylancr 667 . . . . . . 7  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( H  o.  F
) `  X )  =  ( H `  ( F `  X ) ) )
6964, 66, 683eqtr3a 2487 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( G `  X )  =  ( H `  ( F `  X ) ) )
7063, 69oveq12d 6323 . . . . 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 2466 . . . 4  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( H `  ( ( F `  Y )  -  ( F `  X ) ) )  =  ( ( G `
 Y )  -  ( G `  X ) ) )
7271fveq2d 5885 . . 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 4433 . 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 237 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   _Vcvv 3080    C_ wss 3436   class class class wbr 4423    X. cxp 4851   `'ccnv 4852   ran crn 4854    |` cres 4855   "cima 4856    o. ccom 4857    Fn wfn 5596   ` cfv 5601  (class class class)co 6305   RRcr 9545    + caddc 9549   RR*cxr 9681    < clt 9682    - cmin 9867   RR+crp 11309   (,)cioo 11642   abscabs 13297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-ioo 11646  df-seq 12220  df-exp 12279  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299
This theorem is referenced by:  cnre2csqima  28725
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