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Theorem caucvgprlemm 7206
Description: Lemma for caucvgpr 7220. 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 5289 . . . . . 6  |-  ( j  =  1o  ->  ( F `  j )  =  ( F `  1o ) )
21breq2d 3849 . . . . 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 6853 . . . . . 6  |-  1o  e.  N.
54a1i 9 . . . . 5  |-  ( ph  ->  1o  e.  N. )
62, 3, 5rspcdva 2727 . . . 4  |-  ( ph  ->  A  <Q  ( F `  1o ) )
7 ltrelnq 6903 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
87brel 4478 . . . . 5  |-  ( A 
<Q  ( F `  1o )  ->  ( A  e. 
Q.  /\  ( F `  1o )  e.  Q. ) )
98simpld 110 . . . 4  |-  ( A 
<Q  ( F `  1o )  ->  A  e.  Q. )
10 halfnqq 6948 . . . 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 497 . . . . . 6  |-  ( ( ( ph  /\  s  e.  Q. )  /\  (
s  +Q  s )  =  A )  -> 
s  e.  Q. )
13 archrecnq 7201 . . . . . . . 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 108 . . . . . . . . . . . 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 497 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  j  e.  N. )
17 nnnq 6960 . . . . . . . . . . . . . 14  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
18 recclnq 6930 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  s  e.  Q. )
21 ltanqg 6938 . . . . . . . . . . . . 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 1174 . . . . . . . . . . . 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 145 . . . . . . . . . . 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 501 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  (
s  +Q  s )  =  A )
2523, 24breqtrd 3861 . . . . . . . . . 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 2423 . . . . . . . . . . . . 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 478 . . . . . . . . . . 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 6936 . . . . . . . . . . 11  |-  <Q  Or  Q.
3130, 7sotri 4814 . . . . . . . . . 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 403 . . . . . . . . 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 113 . . . . . . . 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 2475 . . . . . . 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 5641 . . . . . . . . 9  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
3736breq1d 3847 . . . . . . . 8  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
3837rexbidv 2381 . . . . . . 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 5292 . . . . . . . 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 6901 . . . . . . . . . 10  |-  Q.  e.  _V
4241rabex 3975 . . . . . . . . 9  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
4341rabex 3975 . . . . . . . . 9  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
4442, 43op1st 5899 . . . . . . . 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 2108 . . . . . . 7  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
4638, 45elrab2 2772 . . . . . 6  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
4712, 35, 46sylanbrc 408 . . . . 5  |-  ( ( ( ph  /\  s  e.  Q. )  /\  (
s  +Q  s )  =  A )  -> 
s  e.  ( 1st `  L ) )
4847ex 113 . . . 4  |-  ( (
ph  /\  s  e.  Q. )  ->  ( ( s  +Q  s )  =  A  ->  s  e.  ( 1st `  L
) ) )
4948reximdva 2475 . . 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 5419 . . . . 5  |-  ( ph  ->  ( F `  1o )  e.  Q. )
53 1nq 6904 . . . . 5  |-  1Q  e.  Q.
54 addclnq 6913 . . . . 5  |-  ( ( ( F `  1o )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( F `  1o )  +Q  1Q )  e. 
Q. )
5552, 53, 54sylancl 404 . . . 4  |-  ( ph  ->  ( ( F `  1o )  +Q  1Q )  e.  Q. )
56 addclnq 6913 . . . 4  |-  ( ( ( ( F `  1o )  +Q  1Q )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( ( F `  1o )  +Q  1Q )  +Q  1Q )  e. 
Q. )
5755, 53, 56sylancl 404 . . 3  |-  ( ph  ->  ( ( ( F `
 1o )  +Q  1Q )  +Q  1Q )  e.  Q. )
58 df-1nqqs 6889 . . . . . . . . 9  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
5958fveq2i 5292 . . . . . . . 8  |-  ( *Q
`  1Q )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
60 rec1nq 6933 . . . . . . . 8  |-  ( *Q
`  1Q )  =  1Q
6159, 60eqtr3i 2110 . . . . . . 7  |-  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  =  1Q
6261oveq2i 5645 . . . . . 6  |-  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )  =  ( ( F `
 1o )  +Q  1Q )
63 ltaddnq 6945 . . . . . . 7  |-  ( ( ( ( F `  1o )  +Q  1Q )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( F `  1o )  +Q  1Q )  <Q 
( ( ( F `
 1o )  +Q  1Q )  +Q  1Q ) )
6455, 53, 63sylancl 404 . . . . . 6  |-  ( ph  ->  ( ( F `  1o )  +Q  1Q )  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
6562, 64syl5eqbr 3870 . . . . 5  |-  ( ph  ->  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
66 opeq1 3617 . . . . . . . . . 10  |-  ( j  =  1o  ->  <. j ,  1o >.  =  <. 1o ,  1o >. )
6766eceq1d 6308 . . . . . . . . 9  |-  ( j  =  1o  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
6867fveq2d 5293 . . . . . . . 8  |-  ( j  =  1o  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )
691, 68oveq12d 5652 . . . . . . 7  |-  ( j  =  1o  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) ) )
7069breq1d 3847 . . . . . 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 2722 . . . . 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 403 . . . 4  |-  ( ph  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
73 breq2 3841 . . . . . 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 2381 . . . . 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 5292 . . . . . 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 5900 . . . . . 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 2108 . . . . 5  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
7874, 77elrab2 2772 . . . 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 408 . . 3  |-  ( ph  ->  ( ( ( F `
 1o )  +Q  1Q )  +Q  1Q )  e.  ( 2nd `  L ) )
80 eleq1 2150 . . . 4  |-  ( r  =  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q )  ->  (
r  e.  ( 2nd `  L )  <->  ( (
( F `  1o )  +Q  1Q )  +Q  1Q )  e.  ( 2nd `  L ) ) )
8180rspcev 2722 . . 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 403 . 2  |-  ( ph  ->  E. r  e.  Q.  r  e.  ( 2nd `  L ) )
8350, 82jca 300 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3444   class class class wbr 3837   -->wf 4998   ` cfv 5002  (class class class)co 5634   1stc1st 5891   2ndc2nd 5892   1oc1o 6156   [cec 6270   N.cnpi 6810    <N clti 6813    ~Q ceq 6817   Q.cnq 6818   1Qc1q 6819    +Q cplq 6820   *Qcrq 6822    <Q cltq 6823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891
This theorem is referenced by:  caucvgprlemcl  7214
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