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Theorem cnre2csqlem 26508
Description: Lemma for cnre2csqima 26509 (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 3487 . . . . . . 7  |-  ( RR 
X.  RR )  C_  _V
3 fnssres 5635 . . . . . . 7  |-  ( ( G  Fn  _V  /\  ( RR  X.  RR )  C_  _V )  -> 
( G  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR ) )
41, 2, 3mp2an 672 . . . . . 6  |-  ( G  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR )
5 elpreima 5935 . . . . . 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 12 . . . . 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 624 . . . 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 434 . . 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 989 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  Y  e.  ( RR  X.  RR ) )
10 fvres 5816 . . . . . 6  |-  ( Y  e.  ( RR  X.  RR )  ->  ( ( G  |`  ( RR  X.  RR ) ) `  Y )  =  ( G `  Y ) )
119, 10syl 16 . . . . 5  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G  |`  ( RR  X.  RR ) ) `
 Y )  =  ( G `  Y
) )
1211eleq1d 2523 . . . 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 988 . . . . . . . . . . . 12  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  X  e.  ( RR  X.  RR ) )
14 fveq2 5802 . . . . . . . . . . . . . 14  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1514eleq1d 2523 . . . . . . . . . . . . 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 3142 . . . . . . . . . . . 12  |-  ( X  e.  ( RR  X.  RR )  ->  ( G `
 X )  e.  RR )
1813, 17syl 16 . . . . . . . . . . 11  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( G `  X )  e.  RR )
19 simp3 990 . . . . . . . . . . . 12  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  D  e.  RR+ )
2019rpred 11142 . . . . . . . . . . 11  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  D  e.  RR )
2118, 20resubcld 9891 . . . . . . . . . 10  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  X
)  -  D )  e.  RR )
2221rexrd 9548 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  X
)  -  D )  e.  RR* )
2318, 20readdcld 9528 . . . . . . . . . 10  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  X
)  +  D )  e.  RR )
2423rexrd 9548 . . . . . . . . 9  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G `  X
)  +  D )  e.  RR* )
25 elioo2 11456 . . . . . . . . 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 661 . . . . . . . 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 484 . . . . . . 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 1001 . . . . . 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 1002 . . . . . 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 532 . . . . 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 434 . . . 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 5802 . . . . . . 7  |-  ( x  =  Y  ->  ( G `  x )  =  ( G `  Y ) )
3433eleq1d 2523 . . . . . 6  |-  ( x  =  Y  ->  (
( G `  x
)  e.  RR  <->  ( G `  Y )  e.  RR ) )
3534, 16vtoclga 3142 . . . . 5  |-  ( Y  e.  ( RR  X.  RR )  ->  ( G `
 Y )  e.  RR )
369, 35syl 16 . . . 4  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( G `  Y )  e.  RR )
37 absdiflt 12927 . . . . 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 1219 . . 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 55 . 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 5952 . . . . . . 7  |-  ( ( F  Fn  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR ) )  -> 
( F `  Y
)  e.  ran  F
)
4341, 9, 42sylancr 663 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( F `  Y )  e.  ran  F )
44 fnfvelrn 5952 . . . . . . 7  |-  ( ( F  Fn  ( RR 
X.  RR )  /\  X  e.  ( RR  X.  RR ) )  -> 
( F `  X
)  e.  ran  F
)
4541, 13, 44sylancr 663 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( F `  X )  e.  ran  F )
46 oveq1 6210 . . . . . . . . 9  |-  ( x  =  ( F `  Y )  ->  (
x  -  y )  =  ( ( F `
 Y )  -  y ) )
4746fveq2d 5806 . . . . . . . 8  |-  ( x  =  ( F `  Y )  ->  ( H `  ( x  -  y ) )  =  ( H `  ( ( F `  Y )  -  y
) ) )
48 fveq2 5802 . . . . . . . . 9  |-  ( x  =  ( F `  Y )  ->  ( H `  x )  =  ( H `  ( F `  Y ) ) )
4948oveq1d 6218 . . . . . . . 8  |-  ( x  =  ( F `  Y )  ->  (
( H `  x
)  -  ( H `
 y ) )  =  ( ( H `
 ( F `  Y ) )  -  ( H `  y ) ) )
5047, 49eqeq12d 2476 . . . . . . 7  |-  ( x  =  ( F `  Y )  ->  (
( H `  (
x  -  y ) )  =  ( ( H `  x )  -  ( H `  y ) )  <->  ( H `  ( ( F `  Y )  -  y
) )  =  ( ( H `  ( F `  Y )
)  -  ( H `
 y ) ) ) )
51 oveq2 6211 . . . . . . . . 9  |-  ( y  =  ( F `  X )  ->  (
( F `  Y
)  -  y )  =  ( ( F `
 Y )  -  ( F `  X ) ) )
5251fveq2d 5806 . . . . . . . 8  |-  ( y  =  ( F `  X )  ->  ( H `  ( ( F `  Y )  -  y ) )  =  ( H `  ( ( F `  Y )  -  ( F `  X )
) ) )
53 fveq2 5802 . . . . . . . . 9  |-  ( y  =  ( F `  X )  ->  ( H `  y )  =  ( H `  ( F `  X ) ) )
5453oveq2d 6219 . . . . . . . 8  |-  ( y  =  ( F `  X )  ->  (
( H `  ( F `  Y )
)  -  ( H `
 y ) )  =  ( ( H `
 ( F `  Y ) )  -  ( H `  ( F `
 X ) ) ) )
5552, 54eqeq12d 2476 . . . . . . 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 3144 . . . . . 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 661 . . . . 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 5803 . . . . . . 7  |-  ( ( G  |`  ( RR  X.  RR ) ) `  Y )  =  ( ( H  o.  F
) `  Y )
61 fvco2 5878 . . . . . . . 8  |-  ( ( F  Fn  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR ) )  -> 
( ( H  o.  F ) `  Y
)  =  ( H `
 ( F `  Y ) ) )
6241, 9, 61sylancr 663 . . . . . . 7  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( H  o.  F
) `  Y )  =  ( H `  ( F `  Y ) ) )
6360, 11, 623eqtr3a 2519 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( G `  Y )  =  ( H `  ( F `  Y ) ) )
6459fveq1i 5803 . . . . . . 7  |-  ( ( G  |`  ( RR  X.  RR ) ) `  X )  =  ( ( H  o.  F
) `  X )
65 fvres 5816 . . . . . . . 8  |-  ( X  e.  ( RR  X.  RR )  ->  ( ( G  |`  ( RR  X.  RR ) ) `  X )  =  ( G `  X ) )
6613, 65syl 16 . . . . . . 7  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( G  |`  ( RR  X.  RR ) ) `
 X )  =  ( G `  X
) )
67 fvco2 5878 . . . . . . . 8  |-  ( ( F  Fn  ( RR 
X.  RR )  /\  X  e.  ( RR  X.  RR ) )  -> 
( ( H  o.  F ) `  X
)  =  ( H `
 ( F `  X ) ) )
6841, 13, 67sylancr 663 . . . . . . 7  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  (
( H  o.  F
) `  X )  =  ( H `  ( F `  X ) ) )
6964, 66, 683eqtr3a 2519 . . . . . 6  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( G `  X )  =  ( H `  ( F `  X ) ) )
7063, 69oveq12d 6221 . . . . 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 2498 . . . 4  |-  ( ( X  e.  ( RR 
X.  RR )  /\  Y  e.  ( RR  X.  RR )  /\  D  e.  RR+ )  ->  ( H `  ( ( F `  Y )  -  ( F `  X ) ) )  =  ( ( G `
 Y )  -  ( G `  X ) ) )
7271fveq2d 5806 . . 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 4413 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3439   class class class wbr 4403    X. cxp 4949   `'ccnv 4950   ran crn 4952    |` cres 4953   "cima 4954    o. ccom 4955    Fn wfn 5524   ` cfv 5529  (class class class)co 6203   RRcr 9396    + caddc 9400   RR*cxr 9532    < clt 9533    - cmin 9710   RR+crp 11106   (,)cioo 11415   abscabs 12845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-ioo 11419  df-seq 11928  df-exp 11987  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847
This theorem is referenced by:  cnre2csqima  26509
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