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Theorem i1faddlem 19048
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 19031 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
31, 2syl 15 . . . . . . . 8  |-  ( ph  ->  F : RR --> RR )
4 ffn 5389 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
53, 4syl 15 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
6 i1fadd.2 . . . . . . . . 9  |-  ( ph  ->  G  e.  dom  S.1 )
7 i1ff 19031 . . . . . . . . 9  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 15 . . . . . . . 8  |-  ( ph  ->  G : RR --> RR )
9 ffn 5389 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
108, 9syl 15 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
11 reex 8828 . . . . . . . 8  |-  RR  e.  _V
1211a1i 10 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
13 inidm 3378 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
145, 10, 12, 12, 13offn 6089 . . . . . 6  |-  ( ph  ->  ( F  o F  +  G )  Fn  RR )
1514adantr 451 . . . . 5  |-  ( (
ph  /\  A  e.  CC )  ->  ( F  o F  +  G
)  Fn  RR )
16 fniniseg 5646 . . . . 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 15 . . . 4  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' ( F  o F  +  G ) " { A } )  <->  ( z  e.  RR  /\  ( ( F  o F  +  G ) `  z
)  =  A ) ) )
1810ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  G  Fn  RR )
19 simprl 732 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  z  e.  RR )
20 fnfvelrn 5662 . . . . . . . 8  |-  ( ( G  Fn  RR  /\  z  e.  RR )  ->  ( G `  z
)  e.  ran  G
)
2118, 19, 20syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  ( G `  z )  e.  ran  G )
22 simprr 733 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  (
( F  o F  +  G ) `  z )  =  A )
23 eqidd 2284 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  RR )  ->  ( F `
 z )  =  ( F `  z
) )
24 eqidd 2284 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  RR )  ->  ( G `
 z )  =  ( G `  z
) )
255, 10, 12, 12, 13, 23, 24ofval 6087 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  RR )  ->  ( ( F  o F  +  G ) `  z
)  =  ( ( F `  z )  +  ( G `  z ) ) )
2625ad2ant2r 727 . . . . . . . . . . . 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 2317 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  A  =  ( ( F `
 z )  +  ( G `  z
) ) )
2827oveq1d 5873 . . . . . . . . . 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 8794 . . . . . . . . . . . . . 14  |-  RR  C_  CC
30 fss 5397 . . . . . . . . . . . . . 14  |-  ( ( F : RR --> RR  /\  RR  C_  CC )  ->  F : RR --> CC )
313, 29, 30sylancl 643 . . . . . . . . . . . . 13  |-  ( ph  ->  F : RR --> CC )
3231ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  F : RR --> CC )
33 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( F : RR --> CC  /\  z  e.  RR )  ->  ( F `  z
)  e.  CC )
3432, 19, 33syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  ( F `  z )  e.  CC )
35 fss 5397 . . . . . . . . . . . . . 14  |-  ( ( G : RR --> RR  /\  RR  C_  CC )  ->  G : RR --> CC )
368, 29, 35sylancl 643 . . . . . . . . . . . . 13  |-  ( ph  ->  G : RR --> CC )
3736ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  G : RR --> CC )
38 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( G : RR --> CC  /\  z  e.  RR )  ->  ( G `  z
)  e.  CC )
3937, 19, 38syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  ( G `  z )  e.  CC )
4034, 39pncand 9158 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  (
( ( F `  z )  +  ( G `  z ) )  -  ( G `
 z ) )  =  ( F `  z ) )
4128, 40eqtr2d 2316 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  ( F `  z )  =  ( A  -  ( G `  z ) ) )
425ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  F  Fn  RR )
43 fniniseg 5646 . . . . . . . . . 10  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { ( A  -  ( G `
 z ) ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  -  ( G `  z )
) ) ) )
4442, 43syl 15 . . . . . . . . 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 )
) ) ) )
4519, 41, 44mpbir2and 888 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  z  e.  ( `' F " { ( A  -  ( G `  z ) ) } ) )
46 eqidd 2284 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  ( G `  z )  =  ( G `  z ) )
47 fniniseg 5646 . . . . . . . . . 10  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { ( G `  z ) } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  ( G `  z
) ) ) )
4818, 47syl 15 . . . . . . . . 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
) ) ) )
4919, 46, 48mpbir2and 888 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  z  e.  ( `' G " { ( G `  z ) } ) )
50 elin 3358 . . . . . . . 8  |-  ( z  e.  ( ( `' F " { ( A  -  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) )  <->  ( z  e.  ( `' F " { ( A  -  ( G `  z ) ) } )  /\  z  e.  ( `' G " { ( G `
 z ) } ) ) )
5145, 49, 50sylanbrc 645 . . . . . . 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 ) } ) ) )
52 oveq2 5866 . . . . . . . . . . . 12  |-  ( y  =  ( G `  z )  ->  ( A  -  y )  =  ( A  -  ( G `  z ) ) )
5352sneqd 3653 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { ( A  -  y ) }  =  { ( A  -  ( G `
 z ) ) } )
5453imaeq2d 5012 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' F " { ( A  -  y ) } )  =  ( `' F " { ( A  -  ( G `
 z ) ) } ) )
55 sneq 3651 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { y }  =  { ( G `  z ) } )
5655imaeq2d 5012 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' G " { y } )  =  ( `' G " { ( G `  z ) } ) )
5754, 56ineq12d 3371 . . . . . . . . 9  |-  ( y  =  ( G `  z )  ->  (
( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  =  ( ( `' F " { ( A  -  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )
5857eleq2d 2350 . . . . . . . 8  |-  ( y  =  ( G `  z )  ->  (
z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  <->  z  e.  ( ( `' F " { ( A  -  ( G `  z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) ) )
5958rspcev 2884 . . . . . . 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 } ) ) )
6021, 51, 59syl2anc 642 . . . . . 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 } ) ) )
6160ex 423 . . . . 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 } ) ) ) )
62 elin 3358 . . . . . . 7  |-  ( z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  <->  ( z  e.  ( `' F " { ( A  -  y ) } )  /\  z  e.  ( `' G " { y } ) ) )
635adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  CC )  ->  F  Fn  RR )
64 fniniseg 5646 . . . . . . . . . 10  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { ( A  -  y ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  -  y
) ) ) )
6563, 64syl 15 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' F " { ( A  -  y ) } )  <-> 
( z  e.  RR  /\  ( F `  z
)  =  ( A  -  y ) ) ) )
6610adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  CC )  ->  G  Fn  RR )
67 fniniseg 5646 . . . . . . . . . 10  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { y } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  y ) ) )
6866, 67syl 15 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( `' G " { y } )  <-> 
( z  e.  RR  /\  ( G `  z
)  =  y ) ) )
6965, 68anbi12d 691 . . . . . . . 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 ) ) ) )
70 anandi 801 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) )  <->  ( (
z  e.  RR  /\  ( F `  z )  =  ( A  -  y ) )  /\  ( z  e.  RR  /\  ( G `  z
)  =  y ) ) )
71 simprl 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
z  e.  RR )
7225ad2ant2r 727 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( F  o F  +  G ) `  z )  =  ( ( F `  z
)  +  ( G `
 z ) ) )
73 simprrl 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( F `  z
)  =  ( A  -  y ) )
74 simprrr 741 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( G `  z
)  =  y )
7573, 74oveq12d 5876 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( F `  z )  +  ( G `  z ) )  =  ( ( A  -  y )  +  y ) )
76 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  ->  A  e.  CC )
7736ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  ->  G : RR --> CC )
7877, 71, 38syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( G `  z
)  e.  CC )
7974, 78eqeltrrd 2358 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
y  e.  CC )
8076, 79npcand 9161 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( A  -  y )  +  y )  =  A )
8172, 75, 803eqtrd 2319 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( F  o F  +  G ) `  z )  =  A )
8271, 81jca 518 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )
8382ex 423 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) )  ->  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) ) )
8470, 83syl5bir 209 . . . . . . . 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 ) ) )
8569, 84sylbid 206 . . . . . . 7  |-  ( (
ph  /\  A  e.  CC )  ->  ( ( z  e.  ( `' F " { ( A  -  y ) } )  /\  z  e.  ( `' G " { y } ) )  ->  ( z  e.  RR  /\  ( ( F  o F  +  G ) `  z
)  =  A ) ) )
8662, 85syl5bi 208 . . . . . 6  |-  ( (
ph  /\  A  e.  CC )  ->  ( z  e.  ( ( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  -> 
( z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) ) )
8786rexlimdvw 2670 . . . . 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 ) ) )
8861, 87impbid 183 . . . 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 } ) ) ) )
8917, 88bitrd 244 . . 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 } ) ) ) )
90 eliun 3909 . . 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 } ) ) )
9189, 90syl6bbr 254 . 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 } ) ) ) )
9291eqrdv 2281 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   {csn 3640   U_ciun 3905   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   RRcr 8736    + caddc 8740    - cmin 9037   S.1citg1 18970
This theorem is referenced by:  i1fadd  19050  itg1addlem4  19054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-sum 12159  df-itg1 18976
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