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Theorem aptiprleml 7629
Description: Lemma for aptipr 7631. (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 7465 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prnmaxl 7478 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  E. s  e.  ( 1st `  A ) x 
<Q  s )
31, 2sylan 283 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  E. s  e.  ( 1st `  A ) x 
<Q  s )
43ad2ant2rl 511 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  E. s  e.  ( 1st `  A
) x  <Q  s
)
5 ltexnqi 7399 . . . . . . 7  |-  ( x 
<Q  s  ->  E. t  e.  Q.  ( x  +Q  t )  =  s )
65ad2antll 491 . . . . . 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 528 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  B  e.  P. )
87ad2antrr 488 . . . . . . . 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 529 . . . . . . . 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 7465 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
11 prarloc2 7494 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  B ) ( u  +Q  t
)  e.  ( 2nd `  B ) )
1210, 11sylan 283 . . . . . . . 8  |-  ( ( B  e.  P.  /\  t  e.  Q. )  ->  E. u  e.  ( 1st `  B ) ( u  +Q  t
)  e.  ( 2nd `  B ) )
138, 9, 12syl2anc 411 . . . . . . 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 276 . . . . . . . . . 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 529 . . . . . . . . . 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 7471 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  u  e.  ( 1st `  B ) )  ->  u  e.  Q. )
1710, 16sylan 283 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  u  e.  ( 1st `  B ) )  ->  u  e.  Q. )
1814, 15, 17syl2anc 411 . . . . . . . . 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 527 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  A  e.  P. )
2019ad3antrrr 492 . . . . . . . . . 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 531 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  x  e.  ( 1st `  A
) )
2221ad3antrrr 492 . . . . . . . . . 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 7471 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
241, 23sylan 283 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
2520, 22, 24syl2anc 411 . . . . . . . . 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 7386 . . . . . . . . 9  |-  ( ( u  e.  Q.  /\  x  e.  Q. )  ->  ( u  <Q  x  \/  u  =  x  \/  x  <Q  u ) )
2718, 25, 26syl2anc 411 . . . . . . . 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 276 . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . 14  |-  ( ( u  e.  Q.  /\  t  e.  Q. )  ->  ( u  +Q  t
)  e.  Q. )
3228, 30, 31syl2anc 411 . . . . . . . . . . . . 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 7390 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
3433adantl 277 . . . . . . . . . . . . . . . . 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 7371 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3635adantl 277 . . . . . . . . . . . . . . . . 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 6038 . . . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . 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 2254 . . . . . . . . . . . . . . . . 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 7474 . . . . . . . . . . . . . . . . . 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 283 . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . 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 150 . . . . . . . . . . . . . . 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 531 . . . . . . . . . . . . . . 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 316 . . . . . . . . . . . . . 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 124 . . . . . . . . . . . . 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 2240 . . . . . . . . . . . . . . 15  |-  ( v  =  ( u  +Q  t )  ->  (
v  e.  ( 2nd `  B )  <->  ( u  +Q  t )  e.  ( 2nd `  B ) ) )
50 eleq1 2240 . . . . . . . . . . . . . . 15  |-  ( v  =  ( u  +Q  t )  ->  (
v  e.  ( 1st `  A )  <->  ( u  +Q  t )  e.  ( 1st `  A ) ) )
5149, 50anbi12d 473 . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 7496 . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 167 . . . . . . . . . . 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 535 . . . . . . . . . . . 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 492 . . . . . . . . . . 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 620 . . . . . . . . . 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 115 . . . . . . . . 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 2240 . . . . . . . . . 10  |-  ( u  =  x  ->  (
u  e.  ( 1st `  B )  <->  x  e.  ( 1st `  B ) ) )
6315, 62syl5ibcom 155 . . . . . . . . 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 7474 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  u  e.  ( 1st `  B ) )  -> 
( x  <Q  u  ->  x  e.  ( 1st `  B ) ) )
6510, 64sylan 283 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  u  e.  ( 1st `  B ) )  -> 
( x  <Q  u  ->  x  e.  ( 1st `  B ) ) )
6614, 15, 65syl2anc 411 . . . . . . . . 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 1304 . . . . . . . 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 2599 . . . . . 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 2599 . . . . 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 2599 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( -.  B  <P  A  /\  x  e.  ( 1st `  A ) ) )  ->  x  e.  ( 1st `  B
) )
7271expr 375 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  -.  B  <P  A )  ->  ( x  e.  ( 1st `  A
)  ->  x  e.  ( 1st `  B ) ) )
73723impa 1194 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( x  e.  ( 1st `  A )  ->  x  e.  ( 1st `  B ) ) )
7473ssrdv 3161 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 104    <-> wb 105    \/ w3o 977    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456    C_ wss 3129   <.cop 3594   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270    +Q cplq 7272    <Q cltq 7275   P.cnp 7281    <P cltp 7285
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iltp 7460
This theorem is referenced by:  aptipr  7631
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