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Theorem i1faddlem 19445
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  o F  +  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 19428 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
31, 2syl 16 . . . . . . . 8  |-  ( ph  ->  F : RR --> RR )
4 ffn 5524 . . . . . . . 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 19428 . . . . . . . . 9  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 16 . . . . . . . 8  |-  ( ph  ->  G : RR --> RR )
9 ffn 5524 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
108, 9syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
11 reex 9007 . . . . . . . 8  |-  RR  e.  _V
1211a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
13 inidm 3486 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
145, 10, 12, 12, 13offn 6248 . . . . . 6  |-  ( ph  ->  ( F  o F  +  G )  Fn  RR )
1514adantr 452 . . . . 5  |-  ( (
ph  /\  A  e.  CC )  ->  ( F  o F  +  G
)  Fn  RR )
16 fniniseg 5783 . . . . 5  |-  ( ( F  o F  +  G )  Fn  RR  ->  ( z  e.  ( `' ( F  o F  +  G ) " { A } )  <-> 
( z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) ) )
1715, 16syl 16 . . . 4  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' ( F  o F  +  G ) " { A } )  <->  ( z  e.  RR  /\  ( ( F  o F  +  G ) `  z
)  =  A ) ) )
1810ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  G  Fn  RR )
19 simprl 733 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  z  e.  RR )
20 fnfvelrn 5799 . . . . . . . 8  |-  ( ( G  Fn  RR  /\  z  e.  RR )  ->  ( G `  z
)  e.  ran  G
)
2118, 19, 20syl2anc 643 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  ( G `  z )  e.  ran  G )
22 simprr 734 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  (
( F  o F  +  G ) `  z )  =  A )
23 eqidd 2381 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  RR )  ->  ( F `
 z )  =  ( F `  z
) )
24 eqidd 2381 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  RR )  ->  ( G `
 z )  =  ( G `  z
) )
255, 10, 12, 12, 13, 23, 24ofval 6246 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  RR )  ->  ( ( F  o F  +  G ) `  z
)  =  ( ( F `  z )  +  ( G `  z ) ) )
2625ad2ant2r 728 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  (
( F  o F  +  G ) `  z )  =  ( ( F `  z
)  +  ( G `
 z ) ) )
2722, 26eqtr3d 2414 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  A  =  ( ( F `
 z )  +  ( G `  z
) ) )
2827oveq1d 6028 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  ( A  -  ( G `  z ) )  =  ( ( ( F `
 z )  +  ( G `  z
) )  -  ( G `  z )
) )
29 ax-resscn 8973 . . . . . . . . . . . . . 14  |-  RR  C_  CC
30 fss 5532 . . . . . . . . . . . . . 14  |-  ( ( F : RR --> RR  /\  RR  C_  CC )  ->  F : RR --> CC )
313, 29, 30sylancl 644 . . . . . . . . . . . . 13  |-  ( ph  ->  F : RR --> CC )
3231ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  F : RR --> CC )
3332, 19ffvelrnd 5803 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  ( F `  z )  e.  CC )
34 fss 5532 . . . . . . . . . . . . . 14  |-  ( ( G : RR --> RR  /\  RR  C_  CC )  ->  G : RR --> CC )
358, 29, 34sylancl 644 . . . . . . . . . . . . 13  |-  ( ph  ->  G : RR --> CC )
3635ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  G : RR --> CC )
3736, 19ffvelrnd 5803 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  ( G `  z )  e.  CC )
3833, 37pncand 9337 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  (
( ( F `  z )  +  ( G `  z ) )  -  ( G `
 z ) )  =  ( F `  z ) )
3928, 38eqtr2d 2413 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  ( F `  z )  =  ( A  -  ( G `  z ) ) )
405ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  F  Fn  RR )
41 fniniseg 5783 . . . . . . . . . 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  o F  +  G ) `  z )  =  A ) )  ->  (
z  e.  ( `' F " { ( A  -  ( G `
 z ) ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  -  ( G `  z )
) ) ) )
4319, 39, 42mpbir2and 889 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  z  e.  ( `' F " { ( A  -  ( G `  z ) ) } ) )
44 eqidd 2381 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  ( G `  z )  =  ( G `  z ) )
45 fniniseg 5783 . . . . . . . . . 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  o F  +  G ) `  z )  =  A ) )  ->  (
z  e.  ( `' G " { ( G `  z ) } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  ( G `  z
) ) ) )
4719, 44, 46mpbir2and 889 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  z  e.  ( `' G " { ( G `  z ) } ) )
48 elin 3466 . . . . . . . 8  |-  ( z  e.  ( ( `' F " { ( A  -  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) )  <->  ( z  e.  ( `' F " { ( A  -  ( G `  z ) ) } )  /\  z  e.  ( `' G " { ( G `
 z ) } ) ) )
4943, 47, 48sylanbrc 646 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  z  e.  ( ( `' F " { ( A  -  ( G `  z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )
50 oveq2 6021 . . . . . . . . . . . 12  |-  ( y  =  ( G `  z )  ->  ( A  -  y )  =  ( A  -  ( G `  z ) ) )
5150sneqd 3763 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { ( A  -  y ) }  =  { ( A  -  ( G `
 z ) ) } )
5251imaeq2d 5136 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' F " { ( A  -  y ) } )  =  ( `' F " { ( A  -  ( G `
 z ) ) } ) )
53 sneq 3761 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { y }  =  { ( G `  z ) } )
5453imaeq2d 5136 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' G " { y } )  =  ( `' G " { ( G `  z ) } ) )
5552, 54ineq12d 3479 . . . . . . . . 9  |-  ( y  =  ( G `  z )  ->  (
( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  =  ( ( `' F " { ( A  -  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )
5655eleq2d 2447 . . . . . . . 8  |-  ( y  =  ( G `  z )  ->  (
z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  <->  z  e.  ( ( `' F " { ( A  -  ( G `  z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) ) )
5756rspcev 2988 . . . . . . 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 } ) ) )
5821, 49, 57syl2anc 643 . . . . . 6  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  E. y  e.  ran  G  z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) )
5958ex 424 . . . . 5  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A )  ->  E. y  e.  ran  G  z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) ) )
60 elin 3466 . . . . . . 7  |-  ( z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  <->  ( z  e.  ( `' F " { ( A  -  y ) } )  /\  z  e.  ( `' G " { y } ) ) )
615adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  CC )  ->  F  Fn  RR )
62 fniniseg 5783 . . . . . . . . . 10  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { ( A  -  y ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  -  y
) ) ) )
6361, 62syl 16 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' F " { ( A  -  y ) } )  <-> 
( z  e.  RR  /\  ( F `  z
)  =  ( A  -  y ) ) ) )
6410adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  CC )  ->  G  Fn  RR )
65 fniniseg 5783 . . . . . . . . . 10  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { y } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  y ) ) )
6664, 65syl 16 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' G " { y } )  <-> 
( z  e.  RR  /\  ( G `  z
)  =  y ) ) )
6763, 66anbi12d 692 . . . . . . . 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 ) ) ) )
68 anandi 802 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) )  <->  ( (
z  e.  RR  /\  ( F `  z )  =  ( A  -  y ) )  /\  ( z  e.  RR  /\  ( G `  z
)  =  y ) ) )
69 simprl 733 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
z  e.  RR )
7025ad2ant2r 728 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( F  o F  +  G ) `  z )  =  ( ( F `  z
)  +  ( G `
 z ) ) )
71 simprrl 741 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( F `  z
)  =  ( A  -  y ) )
72 simprrr 742 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( G `  z
)  =  y )
7371, 72oveq12d 6031 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( F `  z )  +  ( G `  z ) )  =  ( ( A  -  y )  +  y ) )
74 simplr 732 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  ->  A  e.  CC )
7535ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  ->  G : RR --> CC )
7675, 69ffvelrnd 5803 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( G `  z
)  e.  CC )
7772, 76eqeltrrd 2455 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
y  e.  CC )
7874, 77npcand 9340 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( A  -  y )  +  y )  =  A )
7970, 73, 783eqtrd 2416 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( F  o F  +  G ) `  z )  =  A )
8069, 79jca 519 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )
8180ex 424 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) )  ->  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) ) )
8268, 81syl5bir 210 . . . . . . . 8  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( ( z  e.  RR  /\  ( F `  z
)  =  ( A  -  y ) )  /\  ( z  e.  RR  /\  ( G `
 z )  =  y ) )  -> 
( z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) ) )
8367, 82sylbid 207 . . . . . . 7  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( z  e.  ( `' F " { ( A  -  y ) } )  /\  z  e.  ( `' G " { y } ) )  ->  ( z  e.  RR  /\  ( ( F  o F  +  G ) `  z
)  =  A ) ) )
8460, 83syl5bi 209 . . . . . 6  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  -> 
( z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) ) )
8584rexlimdvw 2769 . . . . 5  |-  ( (
ph  /\  A  e.  CC )  ->  ( E. y  e.  ran  G  z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  -> 
( z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) ) )
8659, 85impbid 184 . . . 4  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A )  <->  E. y  e.  ran  G  z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) ) )
8717, 86bitrd 245 . . 3  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' ( F  o F  +  G ) " { A } )  <->  E. y  e.  ran  G  z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) ) )
88 eliun 4032 . . 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 } ) ) )
8987, 88syl6bbr 255 . 2  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' ( F  o F  +  G ) " { A } )  <->  z  e.  U_ y  e.  ran  G
( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) ) )
9089eqrdv 2378 1  |-  ( (
ph  /\  A  e.  CC )  ->  ( `' ( F  o F  +  G ) " { A } )  = 
U_ y  e.  ran  G ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2643   _Vcvv 2892    i^i cin 3255    C_ wss 3256   {csn 3750   U_ciun 4028   `'ccnv 4810   dom cdm 4811   ran crn 4812   "cima 4814    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013    o Fcof 6235   CCcc 8914   RRcr 8915    + caddc 8919    - cmin 9216   S.1citg1 19367
This theorem is referenced by:  i1fadd  19447  itg1addlem4  19451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-ltxr 9051  df-sub 9218  df-sum 12400  df-itg1 19373
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