ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprlemm Unicode version

Theorem caucvgprlemm 7630
Description: Lemma for caucvgpr 7644. 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 5496 . . . . . 6  |-  ( j  =  1o  ->  ( F `  j )  =  ( F `  1o ) )
21breq2d 4001 . . . . 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 7277 . . . . . 6  |-  1o  e.  N.
54a1i 9 . . . . 5  |-  ( ph  ->  1o  e.  N. )
62, 3, 5rspcdva 2839 . . . 4  |-  ( ph  ->  A  <Q  ( F `  1o ) )
7 ltrelnq 7327 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
87brel 4663 . . . . 5  |-  ( A 
<Q  ( F `  1o )  ->  ( A  e. 
Q.  /\  ( F `  1o )  e.  Q. ) )
98simpld 111 . . . 4  |-  ( A 
<Q  ( F `  1o )  ->  A  e.  Q. )
10 halfnqq 7372 . . . 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 525 . . . . . 6  |-  ( ( ( ph  /\  s  e.  Q. )  /\  (
s  +Q  s )  =  A )  -> 
s  e.  Q. )
13 archrecnq 7625 . . . . . . . 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 525 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  j  e.  N. )
17 nnnq 7384 . . . . . . . . . . . . . 14  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
18 recclnq 7354 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  s  e.  Q. )
21 ltanqg 7362 . . . . . . . . . . . . 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 1233 . . . . . . . . . . . 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 529 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  (
s  +Q  s )  =  A )
2523, 24breqtrd 4015 . . . . . . . . . 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 2517 . . . . . . . . . . . . 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 491 . . . . . . . . . . 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 7360 . . . . . . . . . . 11  |-  <Q  Or  Q.
3130, 7sotri 5006 . . . . . . . . . 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 409 . . . . . . . . 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 2572 . . . . . . 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 5860 . . . . . . . . 9  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
3736breq1d 3999 . . . . . . . 8  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
3837rexbidv 2471 . . . . . . 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 5499 . . . . . . . 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 7325 . . . . . . . . . 10  |-  Q.  e.  _V
4241rabex 4133 . . . . . . . . 9  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
4341rabex 4133 . . . . . . . . 9  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
4442, 43op1st 6125 . . . . . . . 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 2191 . . . . . . 7  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
4638, 45elrab2 2889 . . . . . 6  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
4712, 35, 46sylanbrc 415 . . . . 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 2572 . . 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 5632 . . . . 5  |-  ( ph  ->  ( F `  1o )  e.  Q. )
53 1nq 7328 . . . . 5  |-  1Q  e.  Q.
54 addclnq 7337 . . . . 5  |-  ( ( ( F `  1o )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( F `  1o )  +Q  1Q )  e. 
Q. )
5552, 53, 54sylancl 411 . . . 4  |-  ( ph  ->  ( ( F `  1o )  +Q  1Q )  e.  Q. )
56 addclnq 7337 . . . 4  |-  ( ( ( ( F `  1o )  +Q  1Q )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( ( F `  1o )  +Q  1Q )  +Q  1Q )  e. 
Q. )
5755, 53, 56sylancl 411 . . 3  |-  ( ph  ->  ( ( ( F `
 1o )  +Q  1Q )  +Q  1Q )  e.  Q. )
58 df-1nqqs 7313 . . . . . . . . 9  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
5958fveq2i 5499 . . . . . . . 8  |-  ( *Q
`  1Q )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
60 rec1nq 7357 . . . . . . . 8  |-  ( *Q
`  1Q )  =  1Q
6159, 60eqtr3i 2193 . . . . . . 7  |-  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  =  1Q
6261oveq2i 5864 . . . . . 6  |-  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )  =  ( ( F `
 1o )  +Q  1Q )
63 ltaddnq 7369 . . . . . . 7  |-  ( ( ( ( F `  1o )  +Q  1Q )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( F `  1o )  +Q  1Q )  <Q 
( ( ( F `
 1o )  +Q  1Q )  +Q  1Q ) )
6455, 53, 63sylancl 411 . . . . . 6  |-  ( ph  ->  ( ( F `  1o )  +Q  1Q )  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
6562, 64eqbrtrid 4024 . . . . 5  |-  ( ph  ->  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
66 opeq1 3765 . . . . . . . . . 10  |-  ( j  =  1o  ->  <. j ,  1o >.  =  <. 1o ,  1o >. )
6766eceq1d 6549 . . . . . . . . 9  |-  ( j  =  1o  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
6867fveq2d 5500 . . . . . . . 8  |-  ( j  =  1o  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )
691, 68oveq12d 5871 . . . . . . 7  |-  ( j  =  1o  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) ) )
7069breq1d 3999 . . . . . 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 2834 . . . . 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 409 . . . 4  |-  ( ph  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
73 breq2 3993 . . . . . 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 2471 . . . . 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 5499 . . . . . 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 6126 . . . . . 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 2191 . . . . 5  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
7874, 77elrab2 2889 . . . 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 415 . . 3  |-  ( ph  ->  ( ( ( F `
 1o )  +Q  1Q )  +Q  1Q )  e.  ( 2nd `  L ) )
80 eleq1 2233 . . . 4  |-  ( r  =  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q )  ->  (
r  e.  ( 2nd `  L )  <->  ( (
( F `  1o )  +Q  1Q )  +Q  1Q )  e.  ( 2nd `  L ) ) )
8180rspcev 2834 . . 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 409 . 2  |-  ( ph  ->  E. r  e.  Q.  r  e.  ( 2nd `  L ) )
8350, 82jca 304 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 1348    e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   1oc1o 6388   [cec 6511   N.cnpi 7234    <N clti 7237    ~Q ceq 7241   Q.cnq 7242   1Qc1q 7243    +Q cplq 7244   *Qcrq 7246    <Q cltq 7247
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315
This theorem is referenced by:  caucvgprlemcl  7638
  Copyright terms: Public domain W3C validator