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Theorem i1faddlem 19064
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 19047 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
31, 2syl 15 . . . . . . . 8  |-  ( ph  ->  F : RR --> RR )
4 ffn 5405 . . . . . . . 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 19047 . . . . . . . . 9  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 15 . . . . . . . 8  |-  ( ph  ->  G : RR --> RR )
9 ffn 5405 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
108, 9syl 15 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
11 reex 8844 . . . . . . . 8  |-  RR  e.  _V
1211a1i 10 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
13 inidm 3391 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
145, 10, 12, 12, 13offn 6105 . . . . . 6  |-  ( ph  ->  ( F  o F  +  G )  Fn  RR )
1514adantr 451 . . . . 5  |-  ( (
ph  /\  A  e.  CC )  ->  ( F  o F  +  G
)  Fn  RR )
16 fniniseg 5662 . . . . 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 5678 . . . . . . . 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 2297 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  RR )  ->  ( F `
 z )  =  ( F `  z
) )
24 eqidd 2297 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  RR )  ->  ( G `
 z )  =  ( G `  z
) )
255, 10, 12, 12, 13, 23, 24ofval 6103 . . . . . . . . . . . . 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 2330 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  A  =  ( ( F `
 z )  +  ( G `  z
) ) )
2827oveq1d 5889 . . . . . . . . . 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 8810 . . . . . . . . . . . . . 14  |-  RR  C_  CC
30 fss 5413 . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . 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 5413 . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . 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 9174 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  (
( ( F `  z )  +  ( G `  z ) )  -  ( G `
 z ) )  =  ( F `  z ) )
4128, 40eqtr2d 2329 . . . . . . . . 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 5662 . . . . . . . . . 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 2297 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F  o F  +  G ) `  z )  =  A ) )  ->  ( G `  z )  =  ( G `  z ) )
47 fniniseg 5662 . . . . . . . . . 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 3371 . . . . . . . 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 5882 . . . . . . . . . . . 12  |-  ( y  =  ( G `  z )  ->  ( A  -  y )  =  ( A  -  ( G `  z ) ) )
5352sneqd 3666 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { ( A  -  y ) }  =  { ( A  -  ( G `
 z ) ) } )
5453imaeq2d 5028 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' F " { ( A  -  y ) } )  =  ( `' F " { ( A  -  ( G `
 z ) ) } ) )
55 sneq 3664 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { y }  =  { ( G `  z ) } )
5655imaeq2d 5028 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' G " { y } )  =  ( `' G " { ( G `  z ) } ) )
5754, 56ineq12d 3384 . . . . . . . . 9  |-  ( y  =  ( G `  z )  ->  (
( `' F " { ( A  -  y ) } )  i^i  ( `' G " { y } ) )  =  ( ( `' F " { ( A  -  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )
5857eleq2d 2363 . . . . . . . 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 2897 . . . . . . 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 3371 . . . . . . 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 5662 . . . . . . . . . 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 5662 . . . . . . . . . 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 5892 . . . . . . . . . . . 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 2371 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
y  e.  CC )
8076, 79npcand 9177 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  CC )  /\  (
z  e.  RR  /\  ( ( F `  z )  =  ( A  -  y )  /\  ( G `  z )  =  y ) ) )  -> 
( ( A  -  y )  +  y )  =  A )
8172, 75, 803eqtrd 2332 . . . . . . . . . . 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 2683 . . . . 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 3925 . . 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 2294 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   {csn 3653   U_ciun 3921   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   RRcr 8752    + caddc 8756    - cmin 9053   S.1citg1 18986
This theorem is referenced by:  i1fadd  19066  itg1addlem4  19070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-sum 12175  df-itg1 18992
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