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Theorem caucvgprlemm 7658
Description: Lemma for caucvgpr 7672. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.)
Hypotheses
Ref Expression
caucvgpr.f  |-  ( ph  ->  F : N. --> Q. )
caucvgpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
caucvgpr.bnd  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
caucvgpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
Assertion
Ref Expression
caucvgprlemm  |-  ( ph  ->  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  L ) ) )
Distinct variable groups:    A, j, s   
j, F, l    F, r    u, F, j    L, r    ph, j, s    s,
l
Allowed substitution hints:    ph( u, k, n, r, l)    A( u, k, n, r, l)    F( k, n, s)    L( u, j, k, n, s, l)

Proof of Theorem caucvgprlemm
StepHypRef Expression
1 fveq2 5511 . . . . . 6  |-  ( j  =  1o  ->  ( F `  j )  =  ( F `  1o ) )
21breq2d 4012 . . . . 5  |-  ( j  =  1o  ->  ( A  <Q  ( F `  j )  <->  A  <Q  ( F `  1o ) ) )
3 caucvgpr.bnd . . . . 5  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
4 1pi 7305 . . . . . 6  |-  1o  e.  N.
54a1i 9 . . . . 5  |-  ( ph  ->  1o  e.  N. )
62, 3, 5rspcdva 2846 . . . 4  |-  ( ph  ->  A  <Q  ( F `  1o ) )
7 ltrelnq 7355 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
87brel 4675 . . . . 5  |-  ( A 
<Q  ( F `  1o )  ->  ( A  e. 
Q.  /\  ( F `  1o )  e.  Q. ) )
98simpld 112 . . . 4  |-  ( A 
<Q  ( F `  1o )  ->  A  e.  Q. )
10 halfnqq 7400 . . . 4  |-  ( A  e.  Q.  ->  E. s  e.  Q.  ( s  +Q  s )  =  A )
116, 9, 103syl 17 . . 3  |-  ( ph  ->  E. s  e.  Q.  ( s  +Q  s
)  =  A )
12 simplr 528 . . . . . 6  |-  ( ( ( ph  /\  s  e.  Q. )  /\  (
s  +Q  s )  =  A )  -> 
s  e.  Q. )
13 archrecnq 7653 . . . . . . . 8  |-  ( s  e.  Q.  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )
1412, 13syl 14 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  Q. )  /\  (
s  +Q  s )  =  A )  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  s )
15 simpr 110 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  s )
16 simplr 528 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  j  e.  N. )
17 nnnq 7412 . . . . . . . . . . . . . 14  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
18 recclnq 7382 . . . . . . . . . . . . . 14  |-  ( [
<. j ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q. )
1916, 17, 183syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q. )
2012ad2antrr 488 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  s  e.  Q. )
21 ltanqg 7390 . . . . . . . . . . . . 13  |-  ( ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q.  /\  s  e.  Q.  /\  s  e.  Q. )  ->  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( s  +Q  s
) ) )
2219, 20, 20, 21syl3anc 1238 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  (
( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  s  <->  ( s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( s  +Q  s ) ) )
2315, 22mpbid 147 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( s  +Q  s ) )
24 simpllr 534 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  (
s  +Q  s )  =  A )
2523, 24breqtrd 4026 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  A )
26 rsp 2524 . . . . . . . . . . . . 13  |-  ( A. j  e.  N.  A  <Q  ( F `  j
)  ->  ( j  e.  N.  ->  A  <Q  ( F `  j ) ) )
273, 26syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  ( j  e.  N.  ->  A  <Q  ( F `  j ) ) )
2827ad4antr 494 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  (
j  e.  N.  ->  A 
<Q  ( F `  j
) ) )
2916, 28mpd 13 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  A  <Q  ( F `  j
) )
30 ltsonq 7388 . . . . . . . . . . 11  |-  <Q  Or  Q.
3130, 7sotri 5020 . . . . . . . . . 10  |-  ( ( ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  A  /\  A  <Q  ( F `  j
) )  ->  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) )
3225, 29, 31syl2anc 411 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) )
3332ex 115 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  ->  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  <Q  s  ->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
3433reximdva 2579 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  Q. )  /\  (
s  +Q  s )  =  A )  -> 
( E. j  e. 
N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
3514, 34mpd 13 . . . . . 6  |-  ( ( ( ph  /\  s  e.  Q. )  /\  (
s  +Q  s )  =  A )  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) )
36 oveq1 5876 . . . . . . . . 9  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
3736breq1d 4010 . . . . . . . 8  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
3837rexbidv 2478 . . . . . . 7  |-  ( l  =  s  ->  ( E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
39 caucvgpr.lim . . . . . . . . 9  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
4039fveq2i 5514 . . . . . . . 8  |-  ( 1st `  L )  =  ( 1st `  <. { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )
41 nqex 7353 . . . . . . . . . 10  |-  Q.  e.  _V
4241rabex 4144 . . . . . . . . 9  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
4341rabex 4144 . . . . . . . . 9  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
4442, 43op1st 6141 . . . . . . . 8  |-  ( 1st `  <. { l  e. 
Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) } ,  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )  =  { l  e. 
Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) }
4540, 44eqtri 2198 . . . . . . 7  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
4638, 45elrab2 2896 . . . . . 6  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
4712, 35, 46sylanbrc 417 . . . . 5  |-  ( ( ( ph  /\  s  e.  Q. )  /\  (
s  +Q  s )  =  A )  -> 
s  e.  ( 1st `  L ) )
4847ex 115 . . . 4  |-  ( (
ph  /\  s  e.  Q. )  ->  ( ( s  +Q  s )  =  A  ->  s  e.  ( 1st `  L
) ) )
4948reximdva 2579 . . 3  |-  ( ph  ->  ( E. s  e. 
Q.  ( s  +Q  s )  =  A  ->  E. s  e.  Q.  s  e.  ( 1st `  L ) ) )
5011, 49mpd 13 . 2  |-  ( ph  ->  E. s  e.  Q.  s  e.  ( 1st `  L ) )
51 caucvgpr.f . . . . . 6  |-  ( ph  ->  F : N. --> Q. )
5251, 5ffvelcdmd 5648 . . . . 5  |-  ( ph  ->  ( F `  1o )  e.  Q. )
53 1nq 7356 . . . . 5  |-  1Q  e.  Q.
54 addclnq 7365 . . . . 5  |-  ( ( ( F `  1o )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( F `  1o )  +Q  1Q )  e. 
Q. )
5552, 53, 54sylancl 413 . . . 4  |-  ( ph  ->  ( ( F `  1o )  +Q  1Q )  e.  Q. )
56 addclnq 7365 . . . 4  |-  ( ( ( ( F `  1o )  +Q  1Q )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( ( F `  1o )  +Q  1Q )  +Q  1Q )  e. 
Q. )
5755, 53, 56sylancl 413 . . 3  |-  ( ph  ->  ( ( ( F `
 1o )  +Q  1Q )  +Q  1Q )  e.  Q. )
58 df-1nqqs 7341 . . . . . . . . 9  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
5958fveq2i 5514 . . . . . . . 8  |-  ( *Q
`  1Q )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
60 rec1nq 7385 . . . . . . . 8  |-  ( *Q
`  1Q )  =  1Q
6159, 60eqtr3i 2200 . . . . . . 7  |-  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  =  1Q
6261oveq2i 5880 . . . . . 6  |-  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )  =  ( ( F `
 1o )  +Q  1Q )
63 ltaddnq 7397 . . . . . . 7  |-  ( ( ( ( F `  1o )  +Q  1Q )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( F `  1o )  +Q  1Q )  <Q 
( ( ( F `
 1o )  +Q  1Q )  +Q  1Q ) )
6455, 53, 63sylancl 413 . . . . . 6  |-  ( ph  ->  ( ( F `  1o )  +Q  1Q )  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
6562, 64eqbrtrid 4035 . . . . 5  |-  ( ph  ->  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
66 opeq1 3776 . . . . . . . . . 10  |-  ( j  =  1o  ->  <. j ,  1o >.  =  <. 1o ,  1o >. )
6766eceq1d 6565 . . . . . . . . 9  |-  ( j  =  1o  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
6867fveq2d 5515 . . . . . . . 8  |-  ( j  =  1o  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )
691, 68oveq12d 5887 . . . . . . 7  |-  ( j  =  1o  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) ) )
7069breq1d 4010 . . . . . 6  |-  ( j  =  1o  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q )  <->  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )  <Q 
( ( ( F `
 1o )  +Q  1Q )  +Q  1Q ) ) )
7170rspcev 2841 . . . . 5  |-  ( ( 1o  e.  N.  /\  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
725, 65, 71syl2anc 411 . . . 4  |-  ( ph  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
73 breq2 4004 . . . . . 6  |-  ( u  =  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q )  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( ( ( F `
 1o )  +Q  1Q )  +Q  1Q ) ) )
7473rexbidv 2478 . . . . 5  |-  ( u  =  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q )  ->  ( E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( ( ( F `
 1o )  +Q  1Q )  +Q  1Q ) ) )
7539fveq2i 5514 . . . . . 6  |-  ( 2nd `  L )  =  ( 2nd `  <. { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )
7642, 43op2nd 6142 . . . . . 6  |-  ( 2nd `  <. { l  e. 
Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) } ,  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )  =  { u  e. 
Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }
7775, 76eqtri 2198 . . . . 5  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
7874, 77elrab2 2896 . . . 4  |-  ( ( ( ( F `  1o )  +Q  1Q )  +Q  1Q )  e.  ( 2nd `  L
)  <->  ( ( ( ( F `  1o )  +Q  1Q )  +Q  1Q )  e.  Q.  /\ 
E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) ) )
7957, 72, 78sylanbrc 417 . . 3  |-  ( ph  ->  ( ( ( F `
 1o )  +Q  1Q )  +Q  1Q )  e.  ( 2nd `  L ) )
80 eleq1 2240 . . . 4  |-  ( r  =  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q )  ->  (
r  e.  ( 2nd `  L )  <->  ( (
( F `  1o )  +Q  1Q )  +Q  1Q )  e.  ( 2nd `  L ) ) )
8180rspcev 2841 . . 3  |-  ( ( ( ( ( F `
 1o )  +Q  1Q )  +Q  1Q )  e.  Q.  /\  (
( ( F `  1o )  +Q  1Q )  +Q  1Q )  e.  ( 2nd `  L
) )  ->  E. r  e.  Q.  r  e.  ( 2nd `  L ) )
8257, 79, 81syl2anc 411 . 2  |-  ( ph  ->  E. r  e.  Q.  r  e.  ( 2nd `  L ) )
8350, 82jca 306 1  |-  ( ph  ->  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  L ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3594   class class class wbr 4000   -->wf 5208   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   1oc1o 6404   [cec 6527   N.cnpi 7262    <N clti 7265    ~Q ceq 7269   Q.cnq 7270   1Qc1q 7271    +Q cplq 7272   *Qcrq 7274    <Q cltq 7275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343
This theorem is referenced by:  caucvgprlemcl  7666
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