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Theorem aptiprleml 7415
Description: Lemma for aptipr 7417. (Contributed by Jim Kingdon, 28-Jan-2020.)
Assertion
Ref Expression
aptiprleml  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( 1st `  A
)  C_  ( 1st `  B ) )

Proof of Theorem aptiprleml
Dummy variables  f  g  h  s  t  u  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7251 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prnmaxl 7264 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  E. s  e.  ( 1st `  A ) x 
<Q  s )
31, 2sylan 281 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  E. s  e.  ( 1st `  A ) x 
<Q  s )
43ad2ant2rl 502 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  E. s  e.  ( 1st `  A
) x  <Q  s
)
5 ltexnqi 7185 . . . . . . 7  |-  ( x 
<Q  s  ->  E. t  e.  Q.  ( x  +Q  t )  =  s )
65ad2antll 482 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  ( s  e.  ( 1st `  A
)  /\  x  <Q  s ) )  ->  E. t  e.  Q.  ( x  +Q  t )  =  s )
7 simplr 504 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  B  e.  P. )
87ad2antrr 479 . . . . . . . 8  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  ( s  e.  ( 1st `  A
)  /\  x  <Q  s ) )  /\  (
t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  ->  B  e.  P. )
9 simprl 505 . . . . . . . 8  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  ( s  e.  ( 1st `  A
)  /\  x  <Q  s ) )  /\  (
t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  ->  t  e.  Q. )
10 prop 7251 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
11 prarloc2 7280 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  B ) ( u  +Q  t
)  e.  ( 2nd `  B ) )
1210, 11sylan 281 . . . . . . . 8  |-  ( ( B  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  B ) ( u  +Q  t
)  e.  ( 2nd `  B ) )
138, 9, 12syl2anc 408 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  ( s  e.  ( 1st `  A
)  /\  x  <Q  s ) )  /\  (
t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  ->  E. u  e.  ( 1st `  B
) ( u  +Q  t )  e.  ( 2nd `  B ) )
148adantr 274 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  B  e.  P. )
15 simprl 505 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  u  e.  ( 1st `  B
) )
16 elprnql 7257 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  u  e.  ( 1st `  B ) )  ->  u  e.  Q. )
1710, 16sylan 281 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  u  e.  ( 1st `  B ) )  ->  u  e.  Q. )
1814, 15, 17syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  u  e.  Q. )
19 simpll 503 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  A  e.  P. )
2019ad3antrrr 483 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  A  e.  P. )
21 simprr 506 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  x  e.  ( 1st `  A
) )
2221ad3antrrr 483 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  x  e.  ( 1st `  A
) )
23 elprnql 7257 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
241, 23sylan 281 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
2520, 22, 24syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  x  e.  Q. )
26 nqtri3or 7172 . . . . . . . . 9  |-  ( ( u  e.  Q.  /\  x  e.  Q. )  ->  ( u  <Q  x  \/  u  =  x  \/  x  <Q  u ) )
2718, 25, 26syl2anc 408 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  (
u  <Q  x  \/  u  =  x  \/  x  <Q  u ) )
2818adantr 274 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  /\  u  <Q  x )  ->  u  e.  Q. )
29 simplrl 509 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  t  e.  Q. )
3029adantr 274 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  /\  u  <Q  x )  ->  t  e.  Q. )
31 addclnq 7151 . . . . . . . . . . . . . 14  |-  ( ( u  e.  Q.  /\  t  e.  Q. )  ->  ( u  +Q  t
)  e.  Q. )
3228, 30, 31syl2anc 408 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  /\  u  <Q  x )  ->  (
u  +Q  t )  e.  Q. )
33 ltanqg 7176 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
3433adantl 275 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  /\  (
f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. ) )  -> 
( f  <Q  g  <->  ( h  +Q  f ) 
<Q  ( h  +Q  g
) ) )
35 addcomnqg 7157 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3635adantl 275 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  /\  (
f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
3734, 18, 25, 29, 36caovord2d 5908 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  (
u  <Q  x  <->  ( u  +Q  t )  <Q  (
x  +Q  t ) ) )
38 simplrr 510 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  (
x  +Q  t )  =  s )
39 simprl 505 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  ( s  e.  ( 1st `  A
)  /\  x  <Q  s ) )  ->  s  e.  ( 1st `  A
) )
4039ad2antrr 479 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  s  e.  ( 1st `  A
) )
4138, 40eqeltrd 2194 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  (
x  +Q  t )  e.  ( 1st `  A
) )
42 prcdnql 7260 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( x  +Q  t
)  e.  ( 1st `  A ) )  -> 
( ( u  +Q  t )  <Q  (
x  +Q  t )  ->  ( u  +Q  t )  e.  ( 1st `  A ) ) )
431, 42sylan 281 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  P.  /\  ( x  +Q  t
)  e.  ( 1st `  A ) )  -> 
( ( u  +Q  t )  <Q  (
x  +Q  t )  ->  ( u  +Q  t )  e.  ( 1st `  A ) ) )
4420, 41, 43syl2anc 408 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  (
( u  +Q  t
)  <Q  ( x  +Q  t )  ->  (
u  +Q  t )  e.  ( 1st `  A
) ) )
4537, 44sylbid 149 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  (
u  <Q  x  ->  (
u  +Q  t )  e.  ( 1st `  A
) ) )
46 simprr 506 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  (
u  +Q  t )  e.  ( 2nd `  B
) )
4745, 46jctild 314 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  (
u  <Q  x  ->  (
( u  +Q  t
)  e.  ( 2nd `  B )  /\  (
u  +Q  t )  e.  ( 1st `  A
) ) ) )
4847imp 123 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  /\  u  <Q  x )  ->  (
( u  +Q  t
)  e.  ( 2nd `  B )  /\  (
u  +Q  t )  e.  ( 1st `  A
) ) )
49 eleq1 2180 . . . . . . . . . . . . . . 15  |-  ( v  =  ( u  +Q  t )  ->  (
v  e.  ( 2nd `  B )  <->  ( u  +Q  t )  e.  ( 2nd `  B ) ) )
50 eleq1 2180 . . . . . . . . . . . . . . 15  |-  ( v  =  ( u  +Q  t )  ->  (
v  e.  ( 1st `  A )  <->  ( u  +Q  t )  e.  ( 1st `  A ) ) )
5149, 50anbi12d 464 . . . . . . . . . . . . . 14  |-  ( v  =  ( u  +Q  t )  ->  (
( v  e.  ( 2nd `  B )  /\  v  e.  ( 1st `  A ) )  <->  ( ( u  +Q  t )  e.  ( 2nd `  B
)  /\  ( u  +Q  t )  e.  ( 1st `  A ) ) ) )
5251rspcev 2763 . . . . . . . . . . . . 13  |-  ( ( ( u  +Q  t
)  e.  Q.  /\  ( ( u  +Q  t )  e.  ( 2nd `  B )  /\  ( u  +Q  t )  e.  ( 1st `  A ) ) )  ->  E. v  e.  Q.  ( v  e.  ( 2nd `  B
)  /\  v  e.  ( 1st `  A ) ) )
5332, 48, 52syl2anc 408 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  /\  u  <Q  x )  ->  E. v  e.  Q.  ( v  e.  ( 2nd `  B
)  /\  v  e.  ( 1st `  A ) ) )
54 ltdfpr 7282 . . . . . . . . . . . . . 14  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  <P  A  <->  E. v  e.  Q.  ( v  e.  ( 2nd `  B
)  /\  v  e.  ( 1st `  A ) ) ) )
5514, 20, 54syl2anc 408 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  ( B  <P  A  <->  E. v  e.  Q.  ( v  e.  ( 2nd `  B
)  /\  v  e.  ( 1st `  A ) ) ) )
5655adantr 274 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  /\  u  <Q  x )  ->  ( B  <P  A  <->  E. v  e.  Q.  ( v  e.  ( 2nd `  B
)  /\  v  e.  ( 1st `  A ) ) ) )
5753, 56mpbird 166 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  /\  u  <Q  x )  ->  B  <P  A )
58 simplrl 509 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  ( s  e.  ( 1st `  A
)  /\  x  <Q  s ) )  ->  -.  B  <P  A )
5958ad3antrrr 483 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  /\  u  <Q  x )  ->  -.  B  <P  A )
6057, 59pm2.21dd 594 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  /\  u  <Q  x )  ->  x  e.  ( 1st `  B
) )
6160ex 114 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  (
u  <Q  x  ->  x  e.  ( 1st `  B
) ) )
62 eleq1 2180 . . . . . . . . . 10  |-  ( u  =  x  ->  (
u  e.  ( 1st `  B )  <->  x  e.  ( 1st `  B ) ) )
6315, 62syl5ibcom 154 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  (
u  =  x  ->  x  e.  ( 1st `  B ) ) )
64 prcdnql 7260 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  u  e.  ( 1st `  B ) )  -> 
( x  <Q  u  ->  x  e.  ( 1st `  B ) ) )
6510, 64sylan 281 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  u  e.  ( 1st `  B ) )  -> 
( x  <Q  u  ->  x  e.  ( 1st `  B ) ) )
6614, 15, 65syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  (
x  <Q  u  ->  x  e.  ( 1st `  B
) ) )
6761, 63, 663jaod 1267 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  (
( u  <Q  x  \/  u  =  x  \/  x  <Q  u )  ->  x  e.  ( 1st `  B ) ) )
6827, 67mpd 13 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  (
s  e.  ( 1st `  A )  /\  x  <Q  s ) )  /\  ( t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  /\  ( u  e.  ( 1st `  B
)  /\  ( u  +Q  t )  e.  ( 2nd `  B ) ) )  ->  x  e.  ( 1st `  B
) )
6913, 68rexlimddv 2531 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  ( s  e.  ( 1st `  A
)  /\  x  <Q  s ) )  /\  (
t  e.  Q.  /\  ( x  +Q  t
)  =  s ) )  ->  x  e.  ( 1st `  B ) )
706, 69rexlimddv 2531 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  /\  ( s  e.  ( 1st `  A
)  /\  x  <Q  s ) )  ->  x  e.  ( 1st `  B
) )
714, 70rexlimddv 2531 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  x  e.  ( 1st `  B
) )
7271expr 372 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  -.  B  <P  A )  ->  ( x  e.  ( 1st `  A
)  ->  x  e.  ( 1st `  B ) ) )
73723impa 1161 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( x  e.  ( 1st `  A )  ->  x  e.  ( 1st `  B ) ) )
7473ssrdv 3073 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( 1st `  A
)  C_  ( 1st `  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 946    /\ w3a 947    = wceq 1316    e. wcel 1465   E.wrex 2394    C_ wss 3041   <.cop 3500   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056    +Q cplq 7058    <Q cltq 7061   P.cnp 7067    <P cltp 7071
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-2o 6282  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  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iltp 7246
This theorem is referenced by:  aptipr  7417
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