MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolicc2lem2 Unicode version

Theorem ovolicc2lem2 18893
Description: Lemma for ovolicc2 18897. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ovolicc.1  |-  ( ph  ->  A  e.  RR )
ovolicc.2  |-  ( ph  ->  B  e.  RR )
ovolicc.3  |-  ( ph  ->  A  <_  B )
ovolicc2.4  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
ovolicc2.5  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
ovolicc2.6  |-  ( ph  ->  U  e.  ( ~P
ran  ( (,)  o.  F )  i^i  Fin ) )
ovolicc2.7  |-  ( ph  ->  ( A [,] B
)  C_  U. U )
ovolicc2.8  |-  ( ph  ->  G : U --> NN )
ovolicc2.9  |-  ( (
ph  /\  t  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  t ) )  =  t )
ovolicc2.10  |-  T  =  { u  e.  U  |  ( u  i^i  ( A [,] B
) )  =/=  (/) }
ovolicc2.11  |-  ( ph  ->  H : T --> T )
ovolicc2.12  |-  ( (
ph  /\  t  e.  T )  ->  if ( ( 2nd `  ( F `  ( G `  t ) ) )  <_  B ,  ( 2nd `  ( F `
 ( G `  t ) ) ) ,  B )  e.  ( H `  t
) )
ovolicc2.13  |-  ( ph  ->  A  e.  C )
ovolicc2.14  |-  ( ph  ->  C  e.  T )
ovolicc2.15  |-  K  =  seq  1 ( ( H  o.  1st ) ,  ( NN  X.  { C } ) )
ovolicc2.16  |-  W  =  { n  e.  NN  |  B  e.  ( K `  n ) }
Assertion
Ref Expression
ovolicc2lem2  |-  ( (
ph  /\  ( N  e.  NN  /\  -.  N  e.  W ) )  -> 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  <_  B )
Distinct variable groups:    t, n, u, A    B, n, t, u    t, H    C, n, t    n, F, t   
n, K, t, u   
n, G, t    n, W    ph, n, t    T, n, t    n, N, t, u    U, n, t, u
Allowed substitution hints:    ph( u)    C( u)    S( u, t, n)    T( u)    F( u)    G( u)    H( u, n)    W( u, t)

Proof of Theorem ovolicc2lem2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ovolicc.2 . . . . . 6  |-  ( ph  ->  B  e.  RR )
21adantr 451 . . . . 5  |-  ( (
ph  /\  N  e.  NN )  ->  B  e.  RR )
3 ovolicc2.5 . . . . . . . . 9  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
4 inss2 3403 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 fss 5413 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR ) )  ->  F : NN --> ( RR  X.  RR ) )
63, 4, 5sylancl 643 . . . . . . . 8  |-  ( ph  ->  F : NN --> ( RR 
X.  RR ) )
76adantr 451 . . . . . . 7  |-  ( (
ph  /\  N  e.  NN )  ->  F : NN
--> ( RR  X.  RR ) )
8 ovolicc2.8 . . . . . . . . 9  |-  ( ph  ->  G : U --> NN )
98adantr 451 . . . . . . . 8  |-  ( (
ph  /\  N  e.  NN )  ->  G : U
--> NN )
10 nnuz 10279 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
11 ovolicc2.15 . . . . . . . . . . . 12  |-  K  =  seq  1 ( ( H  o.  1st ) ,  ( NN  X.  { C } ) )
12 1z 10069 . . . . . . . . . . . . 13  |-  1  e.  ZZ
1312a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  ZZ )
14 ovolicc2.14 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  T )
15 ovolicc2.11 . . . . . . . . . . . 12  |-  ( ph  ->  H : T --> T )
1610, 11, 13, 14, 15algrf 12759 . . . . . . . . . . 11  |-  ( ph  ->  K : NN --> T )
17 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( K : NN --> T  /\  N  e.  NN )  ->  ( K `  N
)  e.  T )
1816, 17sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  N  e.  NN )  ->  ( K `
 N )  e.  T )
19 ineq1 3376 . . . . . . . . . . . 12  |-  ( u  =  ( K `  N )  ->  (
u  i^i  ( A [,] B ) )  =  ( ( K `  N )  i^i  ( A [,] B ) ) )
2019neeq1d 2472 . . . . . . . . . . 11  |-  ( u  =  ( K `  N )  ->  (
( u  i^i  ( A [,] B ) )  =/=  (/)  <->  ( ( K `
 N )  i^i  ( A [,] B
) )  =/=  (/) ) )
21 ovolicc2.10 . . . . . . . . . . 11  |-  T  =  { u  e.  U  |  ( u  i^i  ( A [,] B
) )  =/=  (/) }
2220, 21elrab2 2938 . . . . . . . . . 10  |-  ( ( K `  N )  e.  T  <->  ( ( K `  N )  e.  U  /\  (
( K `  N
)  i^i  ( A [,] B ) )  =/=  (/) ) )
2318, 22sylib 188 . . . . . . . . 9  |-  ( (
ph  /\  N  e.  NN )  ->  ( ( K `  N )  e.  U  /\  (
( K `  N
)  i^i  ( A [,] B ) )  =/=  (/) ) )
2423simpld 445 . . . . . . . 8  |-  ( (
ph  /\  N  e.  NN )  ->  ( K `
 N )  e.  U )
25 ffvelrn 5679 . . . . . . . 8  |-  ( ( G : U --> NN  /\  ( K `  N )  e.  U )  -> 
( G `  ( K `  N )
)  e.  NN )
269, 24, 25syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  N  e.  NN )  ->  ( G `
 ( K `  N ) )  e.  NN )
27 ffvelrn 5679 . . . . . . 7  |-  ( ( F : NN --> ( RR 
X.  RR )  /\  ( G `  ( K `
 N ) )  e.  NN )  -> 
( F `  ( G `  ( K `  N ) ) )  e.  ( RR  X.  RR ) )
287, 26, 27syl2anc 642 . . . . . 6  |-  ( (
ph  /\  N  e.  NN )  ->  ( F `
 ( G `  ( K `  N ) ) )  e.  ( RR  X.  RR ) )
29 xp2nd 6166 . . . . . 6  |-  ( ( F `  ( G `
 ( K `  N ) ) )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( F `  ( G `  ( K `  N ) ) ) )  e.  RR )
3028, 29syl 15 . . . . 5  |-  ( (
ph  /\  N  e.  NN )  ->  ( 2nd `  ( F `  ( G `  ( K `  N ) ) ) )  e.  RR )
312, 30ltnled 8982 . . . 4  |-  ( (
ph  /\  N  e.  NN )  ->  ( B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  <->  -.  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  <_  B ) )
32 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  N  e.  NN )
331adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  B  e.  RR )
3423adantrr 697 . . . . . . . . . 10  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  (
( K `  N
)  e.  U  /\  ( ( K `  N )  i^i  ( A [,] B ) )  =/=  (/) ) )
3534simprd 449 . . . . . . . . 9  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  (
( K `  N
)  i^i  ( A [,] B ) )  =/=  (/) )
36 n0 3477 . . . . . . . . 9  |-  ( ( ( K `  N
)  i^i  ( A [,] B ) )  =/=  (/) 
<->  E. x  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )
3735, 36sylib 188 . . . . . . . 8  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  E. x  x  e.  ( ( K `  N )  i^i  ( A [,] B
) ) )
38 xp1st 6165 . . . . . . . . . . . . . 14  |-  ( ( F `  ( G `
 ( K `  N ) ) )  e.  ( RR  X.  RR )  ->  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  e.  RR )
3928, 38syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  N  e.  NN )  ->  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  e.  RR )
4039adantrr 697 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  ( 1st `  ( F `  ( G `  ( K `
 N ) ) ) )  e.  RR )
4140adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  e.  RR )
42 simpr 447 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )
43 elin 3371 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( K `
 N )  i^i  ( A [,] B
) )  <->  ( x  e.  ( K `  N
)  /\  x  e.  ( A [,] B ) ) )
4442, 43sylib 188 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( x  e.  ( K `  N
)  /\  x  e.  ( A [,] B ) ) )
4544simprd 449 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  x  e.  ( A [,] B ) )
46 ovolicc.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  RR )
47 elicc2 10731 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4846, 1, 47syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4948ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( x  e.  ( A [,] B
)  <->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) ) )
5045, 49mpbid 201 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
5150simp1d 967 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  x  e.  RR )
521ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  B  e.  RR )
5344simpld 445 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  x  e.  ( K `  N ) )
5434simpld 445 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  ( K `  N )  e.  U )
55 ovolicc.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A  <_  B )
56 ovolicc2.4 . . . . . . . . . . . . . . . 16  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
57 ovolicc2.6 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  U  e.  ( ~P
ran  ( (,)  o.  F )  i^i  Fin ) )
58 ovolicc2.7 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A [,] B
)  C_  U. U )
59 ovolicc2.9 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  t  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  t ) )  =  t )
6046, 1, 55, 56, 3, 57, 58, 8, 59ovolicc2lem1 18892 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( K `  N )  e.  U
)  ->  ( x  e.  ( K `  N
)  <->  ( x  e.  RR  /\  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  x  /\  x  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) ) )
6154, 60syldan 456 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  (
x  e.  ( K `
 N )  <->  ( x  e.  RR  /\  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  x  /\  x  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) ) )
6261adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( x  e.  ( K `  N
)  <->  ( x  e.  RR  /\  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  x  /\  x  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) ) )
6353, 62mpbid 201 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( x  e.  RR  /\  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  x  /\  x  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )
6463simp2d 968 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  x )
6550simp3d 969 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  x  <_  B )
6641, 51, 52, 64, 65ltletrd 8992 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( N  e.  NN  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  /\  x  e.  ( ( K `  N )  i^i  ( A [,] B ) ) )  ->  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  B )
6766ex 423 . . . . . . . . 9  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  (
x  e.  ( ( K `  N )  i^i  ( A [,] B ) )  -> 
( 1st `  ( F `  ( G `  ( K `  N
) ) ) )  <  B ) )
6867exlimdv 1626 . . . . . . . 8  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  ( E. x  x  e.  ( ( K `  N )  i^i  ( A [,] B ) )  ->  ( 1st `  ( F `  ( G `  ( K `  N
) ) ) )  <  B ) )
6937, 68mpd 14 . . . . . . 7  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  ( 1st `  ( F `  ( G `  ( K `
 N ) ) ) )  <  B
)
70 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) )
7146, 1, 55, 56, 3, 57, 58, 8, 59ovolicc2lem1 18892 . . . . . . . 8  |-  ( (
ph  /\  ( K `  N )  e.  U
)  ->  ( B  e.  ( K `  N
)  <->  ( B  e.  RR  /\  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  B  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) ) )
7254, 71syldan 456 . . . . . . 7  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  ( B  e.  ( K `  N )  <->  ( B  e.  RR  /\  ( 1st `  ( F `  ( G `  ( K `  N ) ) ) )  <  B  /\  B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) ) )
7333, 69, 70, 72mpbir3and 1135 . . . . . 6  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  B  e.  ( K `  N
) )
74 fveq2 5541 . . . . . . . 8  |-  ( n  =  N  ->  ( K `  n )  =  ( K `  N ) )
7574eleq2d 2363 . . . . . . 7  |-  ( n  =  N  ->  ( B  e.  ( K `  n )  <->  B  e.  ( K `  N ) ) )
76 ovolicc2.16 . . . . . . 7  |-  W  =  { n  e.  NN  |  B  e.  ( K `  n ) }
7775, 76elrab2 2938 . . . . . 6  |-  ( N  e.  W  <->  ( N  e.  NN  /\  B  e.  ( K `  N
) ) )
7832, 73, 77sylanbrc 645 . . . . 5  |-  ( (
ph  /\  ( N  e.  NN  /\  B  < 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) ) ) )  ->  N  e.  W )
7978expr 598 . . . 4  |-  ( (
ph  /\  N  e.  NN )  ->  ( B  <  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  ->  N  e.  W
) )
8031, 79sylbird 226 . . 3  |-  ( (
ph  /\  N  e.  NN )  ->  ( -.  ( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  <_  B  ->  N  e.  W ) )
8180con1d 116 . 2  |-  ( (
ph  /\  N  e.  NN )  ->  ( -.  N  e.  W  -> 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  <_  B ) )
8281impr 602 1  |-  ( (
ph  /\  ( N  e.  NN  /\  -.  N  e.  W ) )  -> 
( 2nd `  ( F `  ( G `  ( K `  N
) ) ) )  <_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560    i^i cin 3164    C_ wss 3165   (/)c0 3468   ifcif 3578   ~Pcpw 3638   {csn 3653   U.cuni 3843   class class class wbr 4039    X. cxp 4703   ran crn 4706    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Fincfn 6879   RRcr 8752   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   ZZcz 10040   (,)cioo 10672   [,]cicc 10675    seq cseq 11062   abscabs 11735
This theorem is referenced by:  ovolicc2lem3  18894  ovolicc2lem4  18895
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-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  ax-pre-mulgt0 8830
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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-ioo 10676  df-icc 10679  df-fz 10799  df-seq 11063
  Copyright terms: Public domain W3C validator