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Theorem aptiprlemu 6796
Description: Lemma for aptipr 6797. (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 6631 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnminu 6645 . . . . . 6  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  E. s  e.  ( 2nd `  B ) s 
<Q  x )
31, 2sylan 271 . . . . 5  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  E. s  e.  ( 2nd `  B ) s 
<Q  x )
433ad2antl2 1078 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  E. s  e.  ( 2nd `  B
) s  <Q  x
)
5 simprr 492 . . . . . 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 6565 . . . . . 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 918 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  A  e.  P. )
98ad2antrr 465 . . . . . . 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 491 . . . . . . 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 6631 . . . . . . . 8  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 prarloc2 6660 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
1311, 12sylan 271 . . . . . . 7  |-  ( ( A  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
149, 10, 13syl2anc 397 . . . . . 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 919 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  B  e.  P. )
1615ad3antrrr 469 . . . . . . . . 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 107 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  ( 2nd `  B ) )
1817ad3antrrr 469 . . . . . . . . 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 6638 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
201, 19sylan 271 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
2116, 18, 20syl2anc 397 . . . . . . . 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 469 . . . . . . . . . 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 491 . . . . . . . . . 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 6637 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
2511, 24sylan 271 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
2622, 23, 25syl2anc 397 . . . . . . . . 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 265 . . . . . . . . 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 6531 . . . . . . . . 9  |-  ( ( u  e.  Q.  /\  t  e.  Q. )  ->  ( u  +Q  t
)  e.  Q. )
2926, 27, 28syl2anc 397 . . . . . . . 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 6552 . . . . . . . 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 397 . . . . . . 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 265 . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . 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 6638 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  s  e.  ( 2nd `  B ) )  -> 
s  e.  Q. )
351, 34sylan 271 . . . . . . . . . . . . . 14  |-  ( ( B  e.  P.  /\  s  e.  ( 2nd `  B ) )  -> 
s  e.  Q. )
3632, 33, 35syl2anc 397 . . . . . . . . . . . . 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 469 . . . . . . . . . . . 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 469 . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . 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 3795 . . . . . . . . . . . . . . . . 17  |-  ( ( s  +Q  t )  =  x  ->  (
( s  +Q  t
)  <Q  ( u  +Q  t )  <->  x  <Q  ( u  +Q  t ) ) )
4140biimprd 151 . . . . . . . . . . . . . . . 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 119 . . . . . . . . . . . . . 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 6556 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
4544adantl 266 . . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . . 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 6537 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4948adantl 266 . . . . . . . . . . . . . . 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 5698 . . . . . . . . . . . . . 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 160 . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . 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 6640 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  -> 
( s  <Q  u  ->  s  e.  ( 1st `  A ) ) )
5511, 54sylan 271 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  u  e.  ( 1st `  A ) )  -> 
( s  <Q  u  ->  s  e.  ( 1st `  A ) ) )
5652, 53, 55syl2anc 397 . . . . . . . . . . . . 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 2387 . . . . . . . . . . . 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 1144 . . . . . . . . . . 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 265 . . . . . . . . . . . 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 6662 . . . . . . . . . . . 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 397 . . . . . . . . . . 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 160 . . . . . . . . . 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 956 . . . . . . . . . . 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 469 . . . . . . . . . 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 560 . . . . . . . . 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 112 . . . . . . . 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 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 ) ) )  ->  (
u  +Q  t )  e.  ( 2nd `  A
) )
69 eleq1 2116 . . . . . . . . 9  |-  ( x  =  ( u  +Q  t )  ->  (
x  e.  ( 2nd `  A )  <->  ( u  +Q  t )  e.  ( 2nd `  A ) ) )
7068, 69syl5ibrcom 150 . . . . . . . 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 6641 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( u  +Q  t
)  e.  ( 2nd `  A ) )  -> 
( ( u  +Q  t )  <Q  x  ->  x  e.  ( 2nd `  A ) ) )
7211, 71sylan 271 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  ( u  +Q  t
)  e.  ( 2nd `  A ) )  -> 
( ( u  +Q  t )  <Q  x  ->  x  e.  ( 2nd `  A ) ) )
7322, 68, 72syl2anc 397 . . . . . . . 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 1210 . . . . . . 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 2454 . . . . 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 2454 . . . 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 2454 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  ( 2nd `  A ) )
7978ex 112 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( x  e.  ( 2nd `  B )  ->  x  e.  ( 2nd `  A ) ) )
8079ssrdv 2979 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 101    <-> wb 102    \/ w3o 895    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324    C_ wss 2945   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    +Q cplq 6438    <Q cltq 6441   P.cnp 6447    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iltp 6626
This theorem is referenced by:  aptipr  6797
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