MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  i1faddlem Structured version   Unicode version

Theorem i1faddlem 21193
Description: Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1  |-  ( ph  ->  F  e.  dom  S.1 )
i1fadd.2  |-  ( ph  ->  G  e.  dom  S.1 )
Assertion
Ref Expression
i1faddlem  |-  ( (
ph  /\  A  e.  CC )  ->  ( `' ( F  oF  +  G ) " { A } )  = 
U_ y  e.  ran  G ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) )
Distinct variable groups:    y, A    y, F    y, G    ph, y

Proof of Theorem i1faddlem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . . . . . . 9  |-  ( ph  ->  F  e.  dom  S.1 )
2 i1ff 21176 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
31, 2syl 16 . . . . . . . 8  |-  ( ph  ->  F : RR --> RR )
4 ffn 5580 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
53, 4syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
6 i1fadd.2 . . . . . . . . 9  |-  ( ph  ->  G  e.  dom  S.1 )
7 i1ff 21176 . . . . . . . . 9  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 16 . . . . . . . 8  |-  ( ph  ->  G : RR --> RR )
9 ffn 5580 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
108, 9syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
11 reex 9394 . . . . . . . 8  |-  RR  e.  _V
1211a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
13 inidm 3580 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
145, 10, 12, 12, 13offn 6352 . . . . . 6  |-  ( ph  ->  ( F  oF  +  G )  Fn  RR )
1514adantr 465 . . . . 5  |-  ( (
ph  /\  A  e.  CC )  ->  ( F  oF  +  G
)  Fn  RR )
16 fniniseg 5845 . . . . 5  |-  ( ( F  oF  +  G )  Fn  RR  ->  ( z  e.  ( `' ( F  oF  +  G ) " { A } )  <-> 
( z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) ) )
1715, 16syl 16 . . . 4  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' ( F  oF  +  G ) " { A } )  <->  ( z  e.  RR  /\  ( ( F  oF  +  G ) `  z
)  =  A ) ) )
1810ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  G  Fn  RR )
19 simprl 755 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  z  e.  RR )
20 fnfvelrn 5861 . . . . . . . 8  |-  ( ( G  Fn  RR  /\  z  e.  RR )  ->  ( G `  z
)  e.  ran  G
)
2118, 19, 20syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  ( G `  z )  e.  ran  G )
22 simprr 756 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  (
( F  oF  +  G ) `  z )  =  A )
23 eqidd 2444 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  RR )  ->  ( F `
 z )  =  ( F `  z
) )
24 eqidd 2444 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  RR )  ->  ( G `
 z )  =  ( G `  z
) )
255, 10, 12, 12, 13, 23, 24ofval 6350 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  RR )  ->  ( ( F  oF  +  G ) `  z
)  =  ( ( F `  z )  +  ( G `  z ) ) )
2625ad2ant2r 746 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  (
( F  oF  +  G ) `  z )  =  ( ( F `  z
)  +  ( G `
 z ) ) )
2722, 26eqtr3d 2477 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  A  =  ( ( F `
 z )  +  ( G `  z
) ) )
2827oveq1d 6127 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  ( A  -  ( G `  z ) )  =  ( ( ( F `
 z )  +  ( G `  z
) )  -  ( G `  z )
) )
29 ax-resscn 9360 . . . . . . . . . . . . . 14  |-  RR  C_  CC
30 fss 5588 . . . . . . . . . . . . . 14  |-  ( ( F : RR --> RR  /\  RR  C_  CC )  ->  F : RR --> CC )
313, 29, 30sylancl 662 . . . . . . . . . . . . 13  |-  ( ph  ->  F : RR --> CC )
3231ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  F : RR --> CC )
3332, 19ffvelrnd 5865 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  ( F `  z )  e.  CC )
34 fss 5588 . . . . . . . . . . . . . 14  |-  ( ( G : RR --> RR  /\  RR  C_  CC )  ->  G : RR --> CC )
358, 29, 34sylancl 662 . . . . . . . . . . . . 13  |-  ( ph  ->  G : RR --> CC )
3635ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  G : RR --> CC )
3736, 19ffvelrnd 5865 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  ( G `  z )  e.  CC )
3833, 37pncand 9741 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  (
( ( F `  z )  +  ( G `  z ) )  -  ( G `
 z ) )  =  ( F `  z ) )
3928, 38eqtr2d 2476 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  ( F `  z )  =  ( A  -  ( G `  z ) ) )
405ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  F  Fn  RR )
41 fniniseg 5845 . . . . . . . . . 10  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { ( A  -  ( G `
 z ) ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  -  ( G `  z )
) ) ) )
4240, 41syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  (
z  e.  ( `' F " { ( A  -  ( G `
 z ) ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  -  ( G `  z )
) ) ) )
4319, 39, 42mpbir2and 913 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  z  e.  ( `' F " { ( A  -  ( G `  z ) ) } ) )
44 eqidd 2444 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  ( G `  z )  =  ( G `  z ) )
45 fniniseg 5845 . . . . . . . . . 10  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { ( G `  z ) } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  ( G `  z
) ) ) )
4618, 45syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  (
z  e.  ( `' G " { ( G `  z ) } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  ( G `  z
) ) ) )
4719, 44, 46mpbir2and 913 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  z  e.  ( `' G " { ( G `  z ) } ) )
4843, 47elind 3561 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  z  e.  ( ( `' F " { ( A  -  ( G `  z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )
49 oveq2 6120 . . . . . . . . . . . 12  |-  ( y  =  ( G `  z )  ->  ( A  -  y )  =  ( A  -  ( G `  z ) ) )
5049sneqd 3910 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { ( A  -  y ) }  =  { ( A  -  ( G `
 z ) ) } )
5150imaeq2d 5190 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' F " { ( A  -  y ) } )  =  ( `' F " { ( A  -  ( G `
 z ) ) } ) )
52 sneq 3908 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { y }  =  { ( G `  z ) } )
5352imaeq2d 5190 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' G " { y } )  =  ( `' G " { ( G `  z ) } ) )
5451, 53ineq12d 3574 . . . . . . . . 9  |-  ( y  =  ( G `  z )  ->  (
( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  =  ( ( `' F " { ( A  -  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )
5554eleq2d 2510 . . . . . . . 8  |-  ( y  =  ( G `  z )  ->  (
z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  <->  z  e.  ( ( `' F " { ( A  -  ( G `  z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) ) )
5655rspcev 3094 . . . . . . 7  |-  ( ( ( G `  z
)  e.  ran  G  /\  z  e.  (
( `' F " { ( A  -  ( G `  z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )  ->  E. y  e.  ran  G  z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) )
5721, 48, 56syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  E. y  e.  ran  G  z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) )
5857ex 434 . . . . 5  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A )  ->  E. y  e.  ran  G  z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) ) )
59 elin 3560 . . . . . . 7  |-  ( z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  <->  ( z  e.  ( `' F " { ( A  -  y ) } )  /\  z  e.  ( `' G " { y } ) ) )
605adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  CC )  ->  F  Fn  RR )
61 fniniseg 5845 . . . . . . . . . 10  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { ( A  -  y ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  -  y
) ) ) )
6260, 61syl 16 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' F " { ( A  -  y ) } )  <-> 
( z  e.  RR  /\  ( F `  z
)  =  ( A  -  y ) ) ) )
6310adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  CC )  ->  G  Fn  RR )
64 fniniseg 5845 . . . . . . . . . 10  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { y } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  y ) ) )
6563, 64syl 16 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' G " { y } )  <-> 
( z  e.  RR  /\  ( G `  z
)  =  y ) ) )
6662, 65anbi12d 710 . . . . . . . 8  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( z  e.  ( `' F " { ( A  -  y ) } )  /\  z  e.  ( `' G " { y } ) )  <->  ( ( z  e.  RR  /\  ( F `  z )  =  ( A  -  y ) )  /\  ( z  e.  RR  /\  ( G `  z
)  =  y ) ) ) )
67 anandi 824 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) )  <->  ( (
z  e.  RR  /\  ( F `  z )  =  ( A  -  y ) )  /\  ( z  e.  RR  /\  ( G `  z
)  =  y ) ) )
68 simprl 755 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
z  e.  RR )
6925ad2ant2r 746 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( F  oF  +  G ) `  z )  =  ( ( F `  z
)  +  ( G `
 z ) ) )
70 simprrl 763 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( F `  z
)  =  ( A  -  y ) )
71 simprrr 764 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( G `  z
)  =  y )
7270, 71oveq12d 6130 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( F `  z )  +  ( G `  z ) )  =  ( ( A  -  y )  +  y ) )
73 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  ->  A  e.  CC )
7435ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  ->  G : RR --> CC )
7574, 68ffvelrnd 5865 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( G `  z
)  e.  CC )
7671, 75eqeltrrd 2518 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
y  e.  CC )
7773, 76npcand 9744 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( A  -  y )  +  y )  =  A )
7869, 72, 773eqtrd 2479 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( F  oF  +  G ) `  z )  =  A )
7968, 78jca 532 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )
8079ex 434 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) )  ->  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) ) )
8167, 80syl5bir 218 . . . . . . . 8  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( ( z  e.  RR  /\  ( F `  z
)  =  ( A  -  y ) )  /\  ( z  e.  RR  /\  ( G `
 z )  =  y ) )  -> 
( z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) ) )
8266, 81sylbid 215 . . . . . . 7  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( z  e.  ( `' F " { ( A  -  y ) } )  /\  z  e.  ( `' G " { y } ) )  ->  ( z  e.  RR  /\  ( ( F  oF  +  G ) `  z
)  =  A ) ) )
8359, 82syl5bi 217 . . . . . 6  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  -> 
( z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) ) )
8483rexlimdvw 2865 . . . . 5  |-  ( (
ph  /\  A  e.  CC )  ->  ( E. y  e.  ran  G  z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  -> 
( z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) ) )
8558, 84impbid 191 . . . 4  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A )  <->  E. y  e.  ran  G  z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) ) )
8617, 85bitrd 253 . . 3  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' ( F  oF  +  G ) " { A } )  <->  E. y  e.  ran  G  z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) ) )
87 eliun 4196 . . 3  |-  ( z  e.  U_ y  e. 
ran  G ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  <->  E. y  e.  ran  G  z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) )
8886, 87syl6bbr 263 . 2  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' ( F  oF  +  G ) " { A } )  <->  z  e.  U_ y  e.  ran  G
( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) ) )
8988eqrdv 2441 1  |-  ( (
ph  /\  A  e.  CC )  ->  ( `' ( F  oF  +  G ) " { A } )  = 
U_ y  e.  ran  G ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2737   _Vcvv 2993    i^i cin 3348    C_ wss 3349   {csn 3898   U_ciun 4192   `'ccnv 4860   dom cdm 4861   ran crn 4862   "cima 4864    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112    oFcof 6339   CCcc 9301   RRcr 9302    + caddc 9306    - cmin 9616   S.1citg1 21117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-ltxr 9444  df-sub 9618  df-sum 13185  df-itg1 21122
This theorem is referenced by:  i1fadd  21195  itg1addlem4  21199
  Copyright terms: Public domain W3C validator