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Theorem aptiprlemu 7199
Description: Lemma for aptipr 7200. (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 7034 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnminu 7048 . . . . . 6  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  E. s  e.  ( 2nd `  B ) s 
<Q  x )
31, 2sylan 277 . . . . 5  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  E. s  e.  ( 2nd `  B ) s 
<Q  x )
433ad2antl2 1106 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  E. s  e.  ( 2nd `  B
) s  <Q  x
)
5 simprr 499 . . . . . 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 6968 . . . . . 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 946 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  A  e.  P. )
98ad2antrr 472 . . . . . . 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 498 . . . . . . 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 7034 . . . . . . . 8  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
12 prarloc2 7063 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
1311, 12sylan 277 . . . . . . 7  |-  ( ( A  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  A ) ( u  +Q  t
)  e.  ( 2nd `  A ) )
149, 10, 13syl2anc 403 . . . . . 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 947 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  B  e.  P. )
1615ad3antrrr 476 . . . . . . . . 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 108 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  ( 2nd `  B ) )
1817ad3antrrr 476 . . . . . . . . 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 7041 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
201, 19sylan 277 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
2116, 18, 20syl2anc 403 . . . . . . . 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 476 . . . . . . . . . 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 498 . . . . . . . . . 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 7040 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
2511, 24sylan 277 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
2622, 23, 25syl2anc 403 . . . . . . . . 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 270 . . . . . . . . 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 6934 . . . . . . . . 9  |-  ( ( u  e.  Q.  /\  t  e.  Q. )  ->  ( u  +Q  t
)  e.  Q. )
2926, 27, 28syl2anc 403 . . . . . . . 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 6955 . . . . . . . 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 403 . . . . . . 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 270 . . . . . . . . . . . . . 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 498 . . . . . . . . . . . . . 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 7041 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  s  e.  ( 2nd `  B ) )  -> 
s  e.  Q. )
351, 34sylan 277 . . . . . . . . . . . . . 14  |-  ( ( B  e.  P.  /\  s  e.  ( 2nd `  B ) )  -> 
s  e.  Q. )
3632, 33, 35syl2anc 403 . . . . . . . . . . . . 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 476 . . . . . . . . . . . 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 476 . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . . 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 3848 . . . . . . . . . . . . . . . . 17  |-  ( ( s  +Q  t )  =  x  ->  (
( s  +Q  t
)  <Q  ( u  +Q  t )  <->  x  <Q  ( u  +Q  t ) ) )
4140biimprd 156 . . . . . . . . . . . . . . . 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 122 . . . . . . . . . . . . . 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 6959 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
4544adantl 271 . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . 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 6940 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4948adantl 271 . . . . . . . . . . . . . . 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 5814 . . . . . . . . . . . . . 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 165 . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 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 7043 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  -> 
( s  <Q  u  ->  s  e.  ( 1st `  A ) ) )
5511, 54sylan 277 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  u  e.  ( 1st `  A ) )  -> 
( s  <Q  u  ->  s  e.  ( 1st `  A ) ) )
5652, 53, 55syl2anc 403 . . . . . . . . . . . . 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 2424 . . . . . . . . . . . 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 1172 . . . . . . . . . . 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 270 . . . . . . . . . . . 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 7065 . . . . . . . . . . . 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 403 . . . . . . . . . . 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 165 . . . . . . . . . 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 984 . . . . . . . . . . 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 476 . . . . . . . . . 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 585 . . . . . . . . 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 113 . . . . . . . 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 499 . . . . . . . . 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 2150 . . . . . . . . 9  |-  ( x  =  ( u  +Q  t )  ->  (
x  e.  ( 2nd `  A )  <->  ( u  +Q  t )  e.  ( 2nd `  A ) ) )
7068, 69syl5ibrcom 155 . . . . . . . 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 7044 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( u  +Q  t
)  e.  ( 2nd `  A ) )  -> 
( ( u  +Q  t )  <Q  x  ->  x  e.  ( 2nd `  A ) ) )
7211, 71sylan 277 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  ( u  +Q  t
)  e.  ( 2nd `  A ) )  -> 
( ( u  +Q  t )  <Q  x  ->  x  e.  ( 2nd `  A ) ) )
7322, 68, 72syl2anc 403 . . . . . . . 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 1240 . . . . . . 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 2493 . . . . 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 2493 . . . 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 2493 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  ( 2nd `  A ) )
7978ex 113 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( x  e.  ( 2nd `  B )  ->  x  e.  ( 2nd `  A ) ) )
8079ssrdv 3031 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 102    <-> wb 103    \/ w3o 923    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360    C_ wss 2999   <.cop 3449   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   1stc1st 5909   2ndc2nd 5910   Q.cnq 6839    +Q cplq 6841    <Q cltq 6844   P.cnp 6850    <P cltp 6854
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 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
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 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-enq0 6983  df-nq0 6984  df-0nq0 6985  df-plq0 6986  df-mq0 6987  df-inp 7025  df-iltp 7029
This theorem is referenced by:  aptipr  7200
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