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

Theorem caucvgprlemm 7444
Description: Lemma for caucvgpr 7458. 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 5389 . . . . . 6  |-  ( j  =  1o  ->  ( F `  j )  =  ( F `  1o ) )
21breq2d 3911 . . . . 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 7091 . . . . . 6  |-  1o  e.  N.
54a1i 9 . . . . 5  |-  ( ph  ->  1o  e.  N. )
62, 3, 5rspcdva 2768 . . . 4  |-  ( ph  ->  A  <Q  ( F `  1o ) )
7 ltrelnq 7141 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
87brel 4561 . . . . 5  |-  ( A 
<Q  ( F `  1o )  ->  ( A  e. 
Q.  /\  ( F `  1o )  e.  Q. ) )
98simpld 111 . . . 4  |-  ( A 
<Q  ( F `  1o )  ->  A  e.  Q. )
10 halfnqq 7186 . . . 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 504 . . . . . 6  |-  ( ( ( ph  /\  s  e.  Q. )  /\  (
s  +Q  s )  =  A )  -> 
s  e.  Q. )
13 archrecnq 7439 . . . . . . . 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 504 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  j  e.  N. )
17 nnnq 7198 . . . . . . . . . . . . . 14  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
18 recclnq 7168 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  s  e.  Q. )
21 ltanqg 7176 . . . . . . . . . . . . 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 1201 . . . . . . . . . . . 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 508 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e.  Q. )  /\  ( s  +Q  s
)  =  A )  /\  j  e.  N. )  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
s )  ->  (
s  +Q  s )  =  A )
2523, 24breqtrd 3924 . . . . . . . . . 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 2457 . . . . . . . . . . . . 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 485 . . . . . . . . . . 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 7174 . . . . . . . . . . 11  |-  <Q  Or  Q.
3130, 7sotri 4904 . . . . . . . . . 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 408 . . . . . . . . 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 2511 . . . . . . 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 5749 . . . . . . . . 9  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
3736breq1d 3909 . . . . . . . 8  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
3837rexbidv 2415 . . . . . . 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 5392 . . . . . . . 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 7139 . . . . . . . . . 10  |-  Q.  e.  _V
4241rabex 4042 . . . . . . . . 9  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
4341rabex 4042 . . . . . . . . 9  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
4442, 43op1st 6012 . . . . . . . 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 2138 . . . . . . 7  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
4638, 45elrab2 2816 . . . . . 6  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
4712, 35, 46sylanbrc 413 . . . . 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 2511 . . 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 5524 . . . . 5  |-  ( ph  ->  ( F `  1o )  e.  Q. )
53 1nq 7142 . . . . 5  |-  1Q  e.  Q.
54 addclnq 7151 . . . . 5  |-  ( ( ( F `  1o )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( F `  1o )  +Q  1Q )  e. 
Q. )
5552, 53, 54sylancl 409 . . . 4  |-  ( ph  ->  ( ( F `  1o )  +Q  1Q )  e.  Q. )
56 addclnq 7151 . . . 4  |-  ( ( ( ( F `  1o )  +Q  1Q )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( ( F `  1o )  +Q  1Q )  +Q  1Q )  e. 
Q. )
5755, 53, 56sylancl 409 . . 3  |-  ( ph  ->  ( ( ( F `
 1o )  +Q  1Q )  +Q  1Q )  e.  Q. )
58 df-1nqqs 7127 . . . . . . . . 9  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
5958fveq2i 5392 . . . . . . . 8  |-  ( *Q
`  1Q )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
60 rec1nq 7171 . . . . . . . 8  |-  ( *Q
`  1Q )  =  1Q
6159, 60eqtr3i 2140 . . . . . . 7  |-  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  =  1Q
6261oveq2i 5753 . . . . . 6  |-  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )  =  ( ( F `
 1o )  +Q  1Q )
63 ltaddnq 7183 . . . . . . 7  |-  ( ( ( ( F `  1o )  +Q  1Q )  e.  Q.  /\  1Q  e.  Q. )  ->  (
( F `  1o )  +Q  1Q )  <Q 
( ( ( F `
 1o )  +Q  1Q )  +Q  1Q ) )
6455, 53, 63sylancl 409 . . . . . 6  |-  ( ph  ->  ( ( F `  1o )  +Q  1Q )  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
6562, 64eqbrtrid 3933 . . . . 5  |-  ( ph  ->  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
66 opeq1 3675 . . . . . . . . . 10  |-  ( j  =  1o  ->  <. j ,  1o >.  =  <. 1o ,  1o >. )
6766eceq1d 6433 . . . . . . . . 9  |-  ( j  =  1o  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
6867fveq2d 5393 . . . . . . . 8  |-  ( j  =  1o  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )
691, 68oveq12d 5760 . . . . . . 7  |-  ( j  =  1o  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  1o )  +Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) ) )
7069breq1d 3909 . . . . . 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 2763 . . . . 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 408 . . . 4  |-  ( ph  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q ) )
73 breq2 3903 . . . . . 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 2415 . . . . 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 5392 . . . . . 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 6013 . . . . . 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 2138 . . . . 5  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
7874, 77elrab2 2816 . . . 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 413 . . 3  |-  ( ph  ->  ( ( ( F `
 1o )  +Q  1Q )  +Q  1Q )  e.  ( 2nd `  L ) )
80 eleq1 2180 . . . 4  |-  ( r  =  ( ( ( F `  1o )  +Q  1Q )  +Q  1Q )  ->  (
r  e.  ( 2nd `  L )  <->  ( (
( F `  1o )  +Q  1Q )  +Q  1Q )  e.  ( 2nd `  L ) ) )
8180rspcev 2763 . . 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 408 . 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 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   {crab 2397   <.cop 3500   class class class wbr 3899   -->wf 5089   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   1oc1o 6274   [cec 6395   N.cnpi 7048    <N clti 7051    ~Q ceq 7055   Q.cnq 7056   1Qc1q 7057    +Q cplq 7058   *Qcrq 7060    <Q cltq 7061
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129
This theorem is referenced by:  caucvgprlemcl  7452
  Copyright terms: Public domain W3C validator