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Theorem aptiprleml 7440
Description: Lemma for aptipr 7442. (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 7276 . . . . . . 7  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prnmaxl 7289 . . . . . . 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 7210 . . . . . . 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 519 . . . . . . . . 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 520 . . . . . . . 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 7276 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
11 prarloc2 7305 . . . . . . . . 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 520 . . . . . . . . . 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 7282 . . . . . . . . . . 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 518 . . . . . . . . . . 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 521 . . . . . . . . . . 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 7282 . . . . . . . . . . 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 7197 . . . . . . . . 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 524 . . . . . . . . . . . . . . 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 7176 . . . . . . . . . . . . . 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 7201 . . . . . . . . . . . . . . . . . 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 7182 . . . . . . . . . . . . . . . . . 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 5933 . . . . . . . . . . . . . . . 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 525 . . . . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . . . . . 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 2214 . . . . . . . . . . . . . . . . 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 7285 . . . . . . . . . . . . . . . . . 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 521 . . . . . . . . . . . . . . 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 2200 . . . . . . . . . . . . . . 15  |-  ( v  =  ( u  +Q  t )  ->  (
v  e.  ( 2nd `  B )  <->  ( u  +Q  t )  e.  ( 2nd `  B ) ) )
50 eleq1 2200 . . . . . . . . . . . . . . 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 2784 . . . . . . . . . . . . 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 7307 . . . . . . . . . . . . . 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 524 . . . . . . . . . . . 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 609 . . . . . . . . . 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 2200 . . . . . . . . . 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 7285 . . . . . . . . . . 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 1282 . . . . . . . 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 2552 . . . . . 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 2552 . . . . 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 2552 . . . 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 1176 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( x  e.  ( 1st `  A )  ->  x  e.  ( 1st `  B ) ) )
7473ssrdv 3098 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 961    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2415    C_ wss 3066   <.cop 3525   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081    +Q cplq 7083    <Q cltq 7086   P.cnp 7092    <P cltp 7096
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-iltp 7271
This theorem is referenced by:  aptipr  7442
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