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Theorem caucvgprlemm 7440
Description: Lemma for caucvgpr 7454. 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 5387 . . . . . 6  |-  ( j  =  1o  ->  ( F `  j )  =  ( F `  1o ) )
21breq2d 3909 . . . . 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 7087 . . . . . 6  |-  1o  e.  N.
54a1i 9 . . . . 5  |-  ( ph  ->  1o  e.  N. )
62, 3, 5rspcdva 2766 . . . 4  |-  ( ph  ->  A  <Q  ( F `  1o ) )
7 ltrelnq 7137 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
87brel 4559 . . . . 5  |-  ( A 
<Q  ( F `  1o )  ->  ( A  e. 
Q.  /\  ( F `  1o )  e.  Q. ) )
98simpld 111 . . . 4  |-  ( A 
<Q  ( F `  1o )  ->  A  e.  Q. )
10 halfnqq 7182 . . . 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 502 . . . . . 6  |-  ( ( ( ph  /\  s  e.  Q. )  /\  (
s  +Q  s )  =  A )  -> 
s  e.  Q. )
13 archrecnq 7435 . . . . . . . 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 109 . . . . . . . . . . . 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 502 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  j  e.  N. )
17 nnnq 7194 . . . . . . . . . . . . . 14  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
18 recclnq 7164 . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  s  e.  Q. )
21 ltanqg 7172 . . . . . . . . . . . . 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 1199 . . . . . . . . . . . 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 146 . . . . . . . . . . 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 506 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  (
s  +Q  s )  =  A )
2523, 24breqtrd 3922 . . . . . . . . . 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 2455 . . . . . . . . . . . . 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 483 . . . . . . . . . . 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 7170 . . . . . . . . . . 11  |-  <Q  Or  Q.
3130, 7sotri 4902 . . . . . . . . . 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 406 . . . . . . . . 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 114 . . . . . . . 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 2509 . . . . . . 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 5747 . . . . . . . . 9  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
3736breq1d 3907 . . . . . . . 8  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
3837rexbidv 2413 . . . . . . 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 5390 . . . . . . . 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 7135 . . . . . . . . . 10  |-  Q.  e.  _V
4241rabex 4040 . . . . . . . . 9  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
4341rabex 4040 . . . . . . . . 9  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
4442, 43op1st 6010 . . . . . . . 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 2136 . . . . . . 7  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
4638, 45elrab2 2814 . . . . . 6  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
4712, 35, 46sylanbrc 411 . . . . 5  |-  ( ( ( ph  /\  s  e.  Q. )  /\  (
s  +Q  s )  =  A )  -> 
s  e.  ( 1st `  L ) )
4847ex 114 . . . 4  |-  ( (
ph  /\  s  e.  Q. )  ->  ( ( s  +Q  s )  =  A  ->  s  e.  ( 1st `  L
) ) )
4948reximdva 2509 . . 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, 5ffvelrnd 5522 . . . . 5  |-  ( ph  ->  ( F `  1o )  e.  Q. )
53 1nq 7138 . . . . 5  |-  1Q  e.  Q.
54 addclnq 7147 . . . . 5  |-  ( ( ( F `  1o )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( F `  1o )  +Q  1Q )  e. 
Q. )
5552, 53, 54sylancl 407 . . . 4  |-  ( ph  ->  ( ( F `  1o )  +Q  1Q )  e.  Q. )
56 addclnq 7147 . . . 4  |-  ( ( ( ( F `  1o )  +Q  1Q )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( ( F `  1o )  +Q  1Q )  +Q  1Q )  e. 
Q. )
5755, 53, 56sylancl 407 . . 3  |-  ( ph  ->  ( ( ( F `
 1o )  +Q  1Q )  +Q  1Q )  e.  Q. )
58 df-1nqqs 7123 . . . . . . . . 9  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
5958fveq2i 5390 . . . . . . . 8  |-  ( *Q
`  1Q )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
60 rec1nq 7167 . . . . . . . 8  |-  ( *Q
`  1Q )  =  1Q
6159, 60eqtr3i 2138 . . . . . . 7  |-  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  =  1Q
6261oveq2i 5751 . . . . . 6  |-  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )  =  ( ( F `
 1o )  +Q  1Q )
63 ltaddnq 7179 . . . . . . 7  |-  ( ( ( ( F `  1o )  +Q  1Q )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( F `  1o )  +Q  1Q )  <Q 
( ( ( F `
 1o )  +Q  1Q )  +Q  1Q ) )
6455, 53, 63sylancl 407 . . . . . 6  |-  ( ph  ->  ( ( F `  1o )  +Q  1Q )  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
6562, 64eqbrtrid 3931 . . . . 5  |-  ( ph  ->  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
66 opeq1 3673 . . . . . . . . . 10  |-  ( j  =  1o  ->  <. j ,  1o >.  =  <. 1o ,  1o >. )
6766eceq1d 6431 . . . . . . . . 9  |-  ( j  =  1o  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
6867fveq2d 5391 . . . . . . . 8  |-  ( j  =  1o  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )
691, 68oveq12d 5758 . . . . . . 7  |-  ( j  =  1o  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) ) )
7069breq1d 3907 . . . . . 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 2761 . . . . 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 406 . . . 4  |-  ( ph  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
73 breq2 3901 . . . . . 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 2413 . . . . 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 5390 . . . . . 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 6011 . . . . . 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 2136 . . . . 5  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
7874, 77elrab2 2814 . . . 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 411 . . 3  |-  ( ph  ->  ( ( ( F `
 1o )  +Q  1Q )  +Q  1Q )  e.  ( 2nd `  L ) )
80 eleq1 2178 . . . 4  |-  ( r  =  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q )  ->  (
r  e.  ( 2nd `  L )  <->  ( (
( F `  1o )  +Q  1Q )  +Q  1Q )  e.  ( 2nd `  L ) ) )
8180rspcev 2761 . . 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 406 . 2  |-  ( ph  ->  E. r  e.  Q.  r  e.  ( 2nd `  L ) )
8350, 82jca 302 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 103    <-> wb 104    = wceq 1314    e. wcel 1463   A.wral 2391   E.wrex 2392   {crab 2395   <.cop 3498   class class class wbr 3897   -->wf 5087   ` cfv 5091  (class class class)co 5740   1stc1st 6002   2ndc2nd 6003   1oc1o 6272   [cec 6393   N.cnpi 7044    <N clti 7047    ~Q ceq 7051   Q.cnq 7052   1Qc1q 7053    +Q cplq 7054   *Qcrq 7056    <Q cltq 7057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125
This theorem is referenced by:  caucvgprlemcl  7448
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