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Theorem aptiprleml 7572
Description: Lemma for aptipr 7574. (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 7408 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prnmaxl 7421 . . . . . . 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 503 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  E. s  e.  ( 1st `  A
) x  <Q  s
)
5 ltexnqi 7342 . . . . . . 7  |-  ( x 
<Q  s  ->  E. t  e.  Q.  ( x  +Q  t )  =  s )
65ad2antll 483 . . . . . 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 520 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  B  e.  P. )
87ad2antrr 480 . . . . . . . 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 521 . . . . . . . 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 7408 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
11 prarloc2 7437 . . . . . . . . 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 409 . . . . . . 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 521 . . . . . . . . . 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 7414 . . . . . . . . . . 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 409 . . . . . . . . 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 519 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  A  e.  P. )
2019ad3antrrr 484 . . . . . . . . . 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 522 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  x  e.  ( 1st `  A
) )
2221ad3antrrr 484 . . . . . . . . . 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 7414 . . . . . . . . . . 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 409 . . . . . . . . 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 7329 . . . . . . . . 9  |-  ( ( u  e.  Q.  /\  x  e.  Q. )  ->  ( u  <Q  x  \/  u  =  x  \/  x  <Q  u ) )
2718, 25, 26syl2anc 409 . . . . . . . 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 525 . . . . . . . . . . . . . . 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 7308 . . . . . . . . . . . . . 14  |-  ( ( u  e.  Q.  /\  t  e.  Q. )  ->  ( u  +Q  t
)  e.  Q. )
3228, 30, 31syl2anc 409 . . . . . . . . . . . . 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 7333 . . . . . . . . . . . . . . . . . 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 7314 . . . . . . . . . . . . . . . . . 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 6003 . . . . . . . . . . . . . . . 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 526 . . . . . . . . . . . . . . . . . 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 521 . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 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 2241 . . . . . . . . . . . . . . . . 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 7417 . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . 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 522 . . . . . . . . . . . . . . 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 2227 . . . . . . . . . . . . . . 15  |-  ( v  =  ( u  +Q  t )  ->  (
v  e.  ( 2nd `  B )  <->  ( u  +Q  t )  e.  ( 2nd `  B ) ) )
50 eleq1 2227 . . . . . . . . . . . . . . 15  |-  ( v  =  ( u  +Q  t )  ->  (
v  e.  ( 1st `  A )  <->  ( u  +Q  t )  e.  ( 1st `  A ) ) )
5149, 50anbi12d 465 . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . 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 409 . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . 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 525 . . . . . . . . . . . 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 484 . . . . . . . . . . 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 610 . . . . . . . . . 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 2227 . . . . . . . . . 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 7417 . . . . . . . . . . 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 409 . . . . . . . . 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 1293 . . . . . . . 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 2586 . . . . . 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 2586 . . . . 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 2586 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  x  e.  ( 1st `  B
) )
7271expr 373 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  -.  B  <P  A )  ->  ( x  e.  ( 1st `  A
)  ->  x  e.  ( 1st `  B ) ) )
73723impa 1183 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( x  e.  ( 1st `  A )  ->  x  e.  ( 1st `  B ) ) )
7473ssrdv 3144 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 966    /\ w3a 967    = wceq 1342    e. wcel 2135   E.wrex 2443    C_ wss 3112   <.cop 3574   class class class wbr 3977   ` cfv 5183  (class class class)co 5837   1stc1st 6099   2ndc2nd 6100   Q.cnq 7213    +Q cplq 7215    <Q cltq 7218   P.cnp 7224    <P cltp 7228
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4092  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-iinf 4560
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2724  df-sbc 2948  df-csb 3042  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-int 3820  df-iun 3863  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-eprel 4262  df-id 4266  df-po 4269  df-iso 4270  df-iord 4339  df-on 4341  df-suc 4344  df-iom 4563  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190  df-fv 5191  df-ov 5840  df-oprab 5841  df-mpo 5842  df-1st 6101  df-2nd 6102  df-recs 6265  df-irdg 6330  df-1o 6376  df-2o 6377  df-oadd 6380  df-omul 6381  df-er 6493  df-ec 6495  df-qs 6499  df-ni 7237  df-pli 7238  df-mi 7239  df-lti 7240  df-plpq 7277  df-mpq 7278  df-enq 7280  df-nqqs 7281  df-plqqs 7282  df-mqqs 7283  df-1nqqs 7284  df-rq 7285  df-ltnqqs 7286  df-enq0 7357  df-nq0 7358  df-0nq0 7359  df-plq0 7360  df-mq0 7361  df-inp 7399  df-iltp 7403
This theorem is referenced by:  aptipr  7574
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