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Theorem i1faddlem 21015
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 20998 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
31, 2syl 16 . . . . . . . 8  |-  ( ph  ->  F : RR --> RR )
4 ffn 5549 . . . . . . . 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 20998 . . . . . . . . 9  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 16 . . . . . . . 8  |-  ( ph  ->  G : RR --> RR )
9 ffn 5549 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
108, 9syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
11 reex 9363 . . . . . . . 8  |-  RR  e.  _V
1211a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
13 inidm 3549 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
145, 10, 12, 12, 13offn 6322 . . . . . 6  |-  ( ph  ->  ( F  oF  +  G )  Fn  RR )
1514adantr 462 . . . . 5  |-  ( (
ph  /\  A  e.  CC )  ->  ( F  oF  +  G
)  Fn  RR )
16 fniniseg 5814 . . . . 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 720 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  G  Fn  RR )
19 simprl 750 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  z  e.  RR )
20 fnfvelrn 5830 . . . . . . . 8  |-  ( ( G  Fn  RR  /\  z  e.  RR )  ->  ( G `  z
)  e.  ran  G
)
2118, 19, 20syl2anc 656 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  ( G `  z )  e.  ran  G )
22 simprr 751 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  (
( F  oF  +  G ) `  z )  =  A )
23 eqidd 2436 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  RR )  ->  ( F `
 z )  =  ( F `  z
) )
24 eqidd 2436 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  RR )  ->  ( G `
 z )  =  ( G `  z
) )
255, 10, 12, 12, 13, 23, 24ofval 6320 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  RR )  ->  ( ( F  oF  +  G ) `  z
)  =  ( ( F `  z )  +  ( G `  z ) ) )
2625ad2ant2r 741 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  (
( F  oF  +  G ) `  z )  =  ( ( F `  z
)  +  ( G `
 z ) ) )
2722, 26eqtr3d 2469 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  A  =  ( ( F `
 z )  +  ( G `  z
) ) )
2827oveq1d 6097 . . . . . . . . . 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 9329 . . . . . . . . . . . . . 14  |-  RR  C_  CC
30 fss 5557 . . . . . . . . . . . . . 14  |-  ( ( F : RR --> RR  /\  RR  C_  CC )  ->  F : RR --> CC )
313, 29, 30sylancl 657 . . . . . . . . . . . . 13  |-  ( ph  ->  F : RR --> CC )
3231ad2antrr 720 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  F : RR --> CC )
3332, 19ffvelrnd 5834 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  ( F `  z )  e.  CC )
34 fss 5557 . . . . . . . . . . . . . 14  |-  ( ( G : RR --> RR  /\  RR  C_  CC )  ->  G : RR --> CC )
358, 29, 34sylancl 657 . . . . . . . . . . . . 13  |-  ( ph  ->  G : RR --> CC )
3635ad2antrr 720 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  G : RR --> CC )
3736, 19ffvelrnd 5834 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  ( G `  z )  e.  CC )
3833, 37pncand 9710 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  (
( ( F `  z )  +  ( G `  z ) )  -  ( G `
 z ) )  =  ( F `  z ) )
3928, 38eqtr2d 2468 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  ( F `  z )  =  ( A  -  ( G `  z ) ) )
405ad2antrr 720 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  F  Fn  RR )
41 fniniseg 5814 . . . . . . . . . 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 908 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  z  e.  ( `' F " { ( A  -  ( G `  z ) ) } ) )
44 eqidd 2436 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  ( G `  z )  =  ( G `  z ) )
45 fniniseg 5814 . . . . . . . . . 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 908 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  oF  +  G ) `  z )  =  A ) )  ->  z  e.  ( `' G " { ( G `  z ) } ) )
4843, 47elind 3530 . . . . . . 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 6090 . . . . . . . . . . . 12  |-  ( y  =  ( G `  z )  ->  ( A  -  y )  =  ( A  -  ( G `  z ) ) )
5049sneqd 3879 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { ( A  -  y ) }  =  { ( A  -  ( G `
 z ) ) } )
5150imaeq2d 5159 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' F " { ( A  -  y ) } )  =  ( `' F " { ( A  -  ( G `
 z ) ) } ) )
52 sneq 3877 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { y }  =  { ( G `  z ) } )
5352imaeq2d 5159 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' G " { y } )  =  ( `' G " { ( G `  z ) } ) )
5451, 53ineq12d 3543 . . . . . . . . 9  |-  ( y  =  ( G `  z )  ->  (
( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  =  ( ( `' F " { ( A  -  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )
5554eleq2d 2502 . . . . . . . 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 3064 . . . . . . 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 656 . . . . . 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 3529 . . . . . . 7  |-  ( z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  <->  ( z  e.  ( `' F " { ( A  -  y ) } )  /\  z  e.  ( `' G " { y } ) ) )
605adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  CC )  ->  F  Fn  RR )
61 fniniseg 5814 . . . . . . . . . 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 462 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  CC )  ->  G  Fn  RR )
64 fniniseg 5814 . . . . . . . . . 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 705 . . . . . . . 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 819 . . . . . . . . 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 750 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
z  e.  RR )
6925ad2ant2r 741 . . . . . . . . . . . 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 758 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( F `  z
)  =  ( A  -  y ) )
71 simprrr 759 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( G `  z
)  =  y )
7270, 71oveq12d 6100 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( F `  z )  +  ( G `  z ) )  =  ( ( A  -  y )  +  y ) )
73 simplr 749 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  ->  A  e.  CC )
7435ad2antrr 720 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  ->  G : RR --> CC )
7574, 68ffvelrnd 5834 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( G `  z
)  e.  CC )
7671, 75eqeltrrd 2510 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
y  e.  CC )
7773, 76npcand 9713 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( A  -  y )  +  y )  =  A )
7869, 72, 773eqtrd 2471 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( F  oF  +  G ) `  z )  =  A )
7968, 78jca 529 . . . . . . . . . 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 2836 . . . . 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 4165 . . 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 2433 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 1364    e. wcel 1757   E.wrex 2708   _Vcvv 2964    i^i cin 3317    C_ wss 3318   {csn 3867   U_ciun 4161   `'ccnv 4828   dom cdm 4829   ran crn 4830   "cima 4832    Fn wfn 5403   -->wf 5404   ` cfv 5408  (class class class)co 6082    oFcof 6309   CCcc 9270   RRcr 9271    + caddc 9275    - cmin 9585   S.1citg1 20939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-rep 4393  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-op 3874  df-uni 4082  df-iun 4163  df-br 4283  df-opab 4341  df-mpt 4342  df-id 4625  df-po 4630  df-so 4631  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6311  df-er 7091  df-en 7301  df-dom 7302  df-sdom 7303  df-pnf 9410  df-mnf 9411  df-ltxr 9413  df-sub 9587  df-sum 13150  df-itg1 20944
This theorem is referenced by:  i1fadd  21017  itg1addlem4  21021
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