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Theorem aptiprlemu 7614
Description: Lemma for aptipr 7615. (Contributed by Jim Kingdon, 28-Jan-2020.)
Assertion
Ref Expression
aptiprlemu  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( 2nd `  B
)  C_  ( 2nd `  A ) )

Proof of Theorem aptiprlemu
Dummy variables  f  g  h  s  t  u  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7449 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnminu 7463 . . . . . 6  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  E. s  e.  ( 2nd `  B ) s 
<Q  x )
31, 2sylan 283 . . . . 5  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  E. s  e.  ( 2nd `  B ) s 
<Q  x )
433ad2antl2 1160 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  E. s  e.  ( 2nd `  B
) s  <Q  x
)
5 simprr 531 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  -> 
s  <Q  x )
6 ltexnqi 7383 . . . . . 6  |-  ( s 
<Q  x  ->  E. t  e.  Q.  ( s  +Q  t )  =  x )
75, 6syl 14 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  ->  E. t  e.  Q.  ( s  +Q  t
)  =  x )
8 simpl1 1000 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  A  e.  P. )
98ad2antrr 488 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  /\  ( t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  ->  A  e.  P. )
10 simprl 529 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  /\  ( t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  ->  t  e.  Q. )
11 prop 7449 . . . . . . . 8  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 prarloc2 7478 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
1311, 12sylan 283 . . . . . . 7  |-  ( ( A  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
149, 10, 13syl2anc 411 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  /\  ( t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  ->  E. u  e.  ( 1st `  A
) ( u  +Q  t )  e.  ( 2nd `  A ) )
15 simpl2 1001 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  B  e.  P. )
1615ad3antrrr 492 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  B  e.  P. )
17 simpr 110 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  ( 2nd `  B ) )
1817ad3antrrr 492 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  x  e.  ( 2nd `  B
) )
19 elprnqu 7456 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
201, 19sylan 283 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
2116, 18, 20syl2anc 411 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  x  e.  Q. )
228ad3antrrr 492 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  A  e.  P. )
23 simprl 529 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  u  e.  ( 1st `  A
) )
24 elprnql 7455 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
2511, 24sylan 283 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
2622, 23, 25syl2anc 411 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  u  e.  Q. )
2710adantr 276 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  t  e.  Q. )
28 addclnq 7349 . . . . . . . . 9  |-  ( ( u  e.  Q.  /\  t  e.  Q. )  ->  ( u  +Q  t
)  e.  Q. )
2926, 27, 28syl2anc 411 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
u  +Q  t )  e.  Q. )
30 nqtri3or 7370 . . . . . . . 8  |-  ( ( x  e.  Q.  /\  ( u  +Q  t
)  e.  Q. )  ->  ( x  <Q  (
u  +Q  t )  \/  x  =  ( u  +Q  t )  \/  ( u  +Q  t )  <Q  x
) )
3121, 29, 30syl2anc 411 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
x  <Q  ( u  +Q  t )  \/  x  =  ( u  +Q  t )  \/  (
u  +Q  t ) 
<Q  x ) )
3215adantr 276 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  ->  B  e.  P. )
33 simprl 529 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  -> 
s  e.  ( 2nd `  B ) )
34 elprnqu 7456 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  s  e.  ( 2nd `  B ) )  -> 
s  e.  Q. )
351, 34sylan 283 . . . . . . . . . . . . . 14  |-  ( ( B  e.  P.  /\  s  e.  ( 2nd `  B ) )  -> 
s  e.  Q. )
3632, 33, 35syl2anc 411 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  -> 
s  e.  Q. )
3736ad3antrrr 492 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  s  e.  Q. )
3833ad3antrrr 492 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  s  e.  ( 2nd `  B
) )
39 simplrr 536 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
s  +Q  t )  =  x )
40 breq1 4001 . . . . . . . . . . . . . . . . 17  |-  ( ( s  +Q  t )  =  x  ->  (
( s  +Q  t
)  <Q  ( u  +Q  t )  <->  x  <Q  ( u  +Q  t ) ) )
4140biimprd 158 . . . . . . . . . . . . . . . 16  |-  ( ( s  +Q  t )  =  x  ->  (
x  <Q  ( u  +Q  t )  ->  (
s  +Q  t ) 
<Q  ( u  +Q  t
) ) )
4239, 41syl 14 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
x  <Q  ( u  +Q  t )  ->  (
s  +Q  t ) 
<Q  ( u  +Q  t
) ) )
4342imp 124 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  (
s  +Q  t ) 
<Q  ( u  +Q  t
) )
44 ltanqg 7374 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
4544adantl 277 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  /\  ( t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  /\  (
f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. ) )  -> 
( f  <Q  g  <->  ( h  +Q  f ) 
<Q  ( h  +Q  g
) ) )
4626adantr 276 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  u  e.  Q. )
4727adantr 276 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  t  e.  Q. )
48 addcomnqg 7355 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4948adantl 277 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  /\  ( t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  /\  (
f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
5045, 37, 46, 47, 49caovord2d 6034 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  (
s  <Q  u  <->  ( s  +Q  t )  <Q  (
u  +Q  t ) ) )
5143, 50mpbird 167 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  s  <Q  u )
5222adantr 276 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  A  e.  P. )
5323adantr 276 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  u  e.  ( 1st `  A
) )
54 prcdnql 7458 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  -> 
( s  <Q  u  ->  s  e.  ( 1st `  A ) ) )
5511, 54sylan 283 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  u  e.  ( 1st `  A ) )  -> 
( s  <Q  u  ->  s  e.  ( 1st `  A ) ) )
5652, 53, 55syl2anc 411 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  (
s  <Q  u  ->  s  e.  ( 1st `  A
) ) )
5751, 56mpd 13 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  s  e.  ( 1st `  A
) )
58 rspe 2524 . . . . . . . . . . . 12  |-  ( ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B )  /\  s  e.  ( 1st `  A ) ) )  ->  E. s  e.  Q.  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  A ) ) )
5937, 38, 57, 58syl12anc 1236 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  E. s  e.  Q.  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  A ) ) )
6016adantr 276 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  B  e.  P. )
61 ltdfpr 7480 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  <P  A  <->  E. s  e.  Q.  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  A ) ) ) )
6260, 52, 61syl2anc 411 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  ( B  <P  A  <->  E. s  e.  Q.  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  A ) ) ) )
6359, 62mpbird 167 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  B  <P  A )
64 simpll3 1038 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  ->  -.  B  <P  A )
6564ad3antrrr 492 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  -.  B  <P  A )
6663, 65pm2.21dd 620 . . . . . . . . 9  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B
)  /\  s  <Q  x ) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  /\  x  <Q  ( u  +Q  t
) )  ->  x  e.  ( 2nd `  A
) )
6766ex 115 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
x  <Q  ( u  +Q  t )  ->  x  e.  ( 2nd `  A
) ) )
68 simprr 531 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
u  +Q  t )  e.  ( 2nd `  A
) )
69 eleq1 2238 . . . . . . . . 9  |-  ( x  =  ( u  +Q  t )  ->  (
x  e.  ( 2nd `  A )  <->  ( u  +Q  t )  e.  ( 2nd `  A ) ) )
7068, 69syl5ibrcom 157 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
x  =  ( u  +Q  t )  ->  x  e.  ( 2nd `  A ) ) )
71 prcunqu 7459 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( u  +Q  t
)  e.  ( 2nd `  A ) )  -> 
( ( u  +Q  t )  <Q  x  ->  x  e.  ( 2nd `  A ) ) )
7211, 71sylan 283 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  ( u  +Q  t
)  e.  ( 2nd `  A ) )  -> 
( ( u  +Q  t )  <Q  x  ->  x  e.  ( 2nd `  A ) ) )
7322, 68, 72syl2anc 411 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
( u  +Q  t
)  <Q  x  ->  x  e.  ( 2nd `  A
) ) )
7467, 70, 733jaod 1304 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  (
( x  <Q  (
u  +Q  t )  \/  x  =  ( u  +Q  t )  \/  ( u  +Q  t )  <Q  x
)  ->  x  e.  ( 2nd `  A ) ) )
7531, 74mpd 13 . . . . . 6  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  /\  ( s  e.  ( 2nd `  B )  /\  s  <Q  x
) )  /\  (
t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  /\  ( u  e.  ( 1st `  A
)  /\  ( u  +Q  t )  e.  ( 2nd `  A ) ) )  ->  x  e.  ( 2nd `  A
) )
7614, 75rexlimddv 2597 . . . . 5  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  /\  ( t  e.  Q.  /\  ( s  +Q  t
)  =  x ) )  ->  x  e.  ( 2nd `  A ) )
777, 76rexlimddv 2597 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B
) )  /\  (
s  e.  ( 2nd `  B )  /\  s  <Q  x ) )  ->  x  e.  ( 2nd `  A ) )
784, 77rexlimddv 2597 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  ( 2nd `  A ) )
7978ex 115 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( x  e.  ( 2nd `  B )  ->  x  e.  ( 2nd `  A ) ) )
8079ssrdv 3159 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( 2nd `  B
)  C_  ( 2nd `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ w3o 977    /\ w3a 978    = wceq 1353    e. wcel 2146   E.wrex 2454    C_ wss 3127   <.cop 3592   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   1stc1st 6129   2ndc2nd 6130   Q.cnq 7254    +Q cplq 7256    <Q cltq 7259   P.cnp 7265    <P cltp 7269
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-2o 6408  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-enq0 7398  df-nq0 7399  df-0nq0 7400  df-plq0 7401  df-mq0 7402  df-inp 7440  df-iltp 7444
This theorem is referenced by:  aptipr  7615  suplocexprlemmu  7692
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