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Theorem ltexprlemrl 7530
Description: Lemma for ltexpri 7533. Reverse direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
Assertion
Ref Expression
ltexprlemrl  |-  ( A 
<P  B  ->  ( 1st `  B )  C_  ( 1st `  ( A  +P.  C ) ) )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y

Proof of Theorem ltexprlemrl
Dummy variables  z  w  u  v  f  g  h  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 7425 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
21brel 4638 . . . . . . 7  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simprd 113 . . . . . 6  |-  ( A 
<P  B  ->  B  e. 
P. )
4 prop 7395 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
53, 4syl 14 . . . . 5  |-  ( A 
<P  B  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
6 prnmaddl 7410 . . . . 5  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  w  e.  ( 1st `  B ) )  ->  E. v  e.  Q.  ( w  +Q  v
)  e.  ( 1st `  B ) )
75, 6sylan 281 . . . 4  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  ->  E. v  e.  Q.  ( w  +Q  v
)  e.  ( 1st `  B ) )
82simpld 111 . . . . . . . 8  |-  ( A 
<P  B  ->  A  e. 
P. )
9 prop 7395 . . . . . . . 8  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
108, 9syl 14 . . . . . . 7  |-  ( A 
<P  B  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
11 prarloc 7423 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  v  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( z  +Q  v ) )
1210, 11sylan 281 . . . . . 6  |-  ( ( A  <P  B  /\  v  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( z  +Q  v ) )
1312ad2ant2r 501 . . . . 5  |-  ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  ->  E. z  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( z  +Q  v ) )
14 simplll 523 . . . . . . . . . . 11  |-  ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  (
w  +Q  v )  e.  ( 1st `  B
) ) )  /\  ( z  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A ) ) )  ->  A  <P  B )
1514adantr 274 . . . . . . . . . 10  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  A  <P  B )
16 simplrl 525 . . . . . . . . . 10  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  z  e.  ( 1st `  A
) )
17 elprnql 7401 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
1810, 17sylan 281 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
1915, 16, 18syl2anc 409 . . . . . . . . 9  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  z  e.  Q. )
20 elprnql 7401 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  w  e.  ( 1st `  B ) )  ->  w  e.  Q. )
215, 20sylan 281 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  ->  w  e.  Q. )
2221ad3antrrr 484 . . . . . . . . 9  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  w  e.  Q. )
23 nqtri3or 7316 . . . . . . . . 9  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  <Q  w  \/  z  =  w  \/  w  <Q  z ) )
2419, 22, 23syl2anc 409 . . . . . . . 8  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  (
z  <Q  w  \/  z  =  w  \/  w  <Q  z ) )
25 ltexnqq 7328 . . . . . . . . . . . . 13  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  <Q  w  <->  E. s  e.  Q.  (
z  +Q  s )  =  w ) )
2619, 22, 25syl2anc 409 . . . . . . . . . . . 12  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  (
z  <Q  w  <->  E. s  e.  Q.  ( z  +Q  s )  =  w ) )
2726biimpa 294 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  ->  E. s  e.  Q.  ( z  +Q  s )  =  w )
28 simprr 522 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  ( z  +Q  s )  =  w )
2916ad2antrr 480 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  z  e.  ( 1st `  A ) )
30 simprl 521 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  s  e.  Q. )
31 simpr 109 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  u  <Q  ( z  +Q  v
) )
32 simplrr 526 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  u  e.  ( 2nd `  A
) )
33 prcunqu 7405 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 2nd `  A ) )  -> 
( u  <Q  (
z  +Q  v )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
3410, 33sylan 281 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  <P  B  /\  u  e.  ( 2nd `  A ) )  -> 
( u  <Q  (
z  +Q  v )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
3515, 32, 34syl2anc 409 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  (
u  <Q  ( z  +Q  v )  ->  (
z  +Q  v )  e.  ( 2nd `  A
) ) )
3631, 35mpd 13 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  (
z  +Q  v )  e.  ( 2nd `  A
) )
3736ad2antrr 480 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) )
3819ad2antrr 480 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  z  e.  Q. )
39 simplrl 525 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  (
w  +Q  v )  e.  ( 1st `  B
) ) )  /\  ( z  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A ) ) )  ->  v  e.  Q. )
4039ad3antrrr 484 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  v  e.  Q. )
41 addcomnqg 7301 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4241adantl 275 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  (
w  +Q  v )  e.  ( 1st `  B
) ) )  /\  ( z  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  /\  ( f  e.  Q.  /\  g  e.  Q. ) )  -> 
( f  +Q  g
)  =  ( g  +Q  f ) )
43 addassnqg 7302 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
4443adantl 275 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  (
w  +Q  v )  e.  ( 1st `  B
) ) )  /\  ( z  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e. 
Q. ) )  -> 
( ( f  +Q  g )  +Q  h
)  =  ( f  +Q  ( g  +Q  h ) ) )
4538, 40, 30, 42, 44caov32d 6001 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  ( (
z  +Q  v )  +Q  s )  =  ( ( z  +Q  s )  +Q  v
) )
46 simplrr 526 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  (
w  +Q  v )  e.  ( 1st `  B
) ) )  /\  ( z  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A ) ) )  ->  (
w  +Q  v )  e.  ( 1st `  B
) )
4746ad3antrrr 484 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  ( w  +Q  v )  e.  ( 1st `  B ) )
48 oveq1 5831 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  +Q  s )  =  w  ->  (
( z  +Q  s
)  +Q  v )  =  ( w  +Q  v ) )
4948eleq1d 2226 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  +Q  s )  =  w  ->  (
( ( z  +Q  s )  +Q  v
)  e.  ( 1st `  B )  <->  ( w  +Q  v )  e.  ( 1st `  B ) ) )
5028, 49syl 14 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  ( (
( z  +Q  s
)  +Q  v )  e.  ( 1st `  B
)  <->  ( w  +Q  v )  e.  ( 1st `  B ) ) )
5147, 50mpbird 166 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  ( (
z  +Q  s )  +Q  v )  e.  ( 1st `  B
) )
5245, 51eqeltrd 2234 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  ( (
z  +Q  v )  +Q  s )  e.  ( 1st `  B
) )
53 eleq1 2220 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( z  +Q  v )  ->  (
y  e.  ( 2nd `  A )  <->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
54 oveq1 5831 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( z  +Q  v )  ->  (
y  +Q  s )  =  ( ( z  +Q  v )  +Q  s ) )
5554eleq1d 2226 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( z  +Q  v )  ->  (
( y  +Q  s
)  e.  ( 1st `  B )  <->  ( (
z  +Q  v )  +Q  s )  e.  ( 1st `  B
) ) )
5653, 55anbi12d 465 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( z  +Q  v )  ->  (
( y  e.  ( 2nd `  A )  /\  ( y  +Q  s )  e.  ( 1st `  B ) )  <->  ( ( z  +Q  v )  e.  ( 2nd `  A
)  /\  ( (
z  +Q  v )  +Q  s )  e.  ( 1st `  B
) ) ) )
5756spcegv 2800 . . . . . . . . . . . . . . . 16  |-  ( ( z  +Q  v )  e.  ( 2nd `  A
)  ->  ( (
( z  +Q  v
)  e.  ( 2nd `  A )  /\  (
( z  +Q  v
)  +Q  s )  e.  ( 1st `  B
) )  ->  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  s )  e.  ( 1st `  B ) ) ) )
5857anabsi5 569 . . . . . . . . . . . . . . 15  |-  ( ( ( z  +Q  v
)  e.  ( 2nd `  A )  /\  (
( z  +Q  v
)  +Q  s )  e.  ( 1st `  B
) )  ->  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  s )  e.  ( 1st `  B ) ) )
5937, 52, 58syl2anc 409 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  s )  e.  ( 1st `  B ) ) )
60 ltexprlem.1 . . . . . . . . . . . . . . 15  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
6160ltexprlemell 7518 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 1st `  C
)  <->  ( s  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  s )  e.  ( 1st `  B ) ) ) )
6230, 59, 61sylanbrc 414 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  s  e.  ( 1st `  C ) )
6315, 8syl 14 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  A  e.  P. )
6463ad2antrr 480 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  A  e.  P. )
6560ltexprlempr 7528 . . . . . . . . . . . . . . . 16  |-  ( A 
<P  B  ->  C  e. 
P. )
6615, 65syl 14 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  C  e.  P. )
6766ad2antrr 480 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  C  e.  P. )
68 df-iplp 7388 . . . . . . . . . . . . . . 15  |-  +P.  =  ( x  e.  P. ,  w  e.  P.  |->  <. { z  e.  Q.  |  E. f  e.  Q.  E. v  e.  Q.  (
f  e.  ( 1st `  x )  /\  v  e.  ( 1st `  w
)  /\  z  =  ( f  +Q  v
) ) } ,  { z  e.  Q.  |  E. f  e.  Q.  E. v  e.  Q.  (
f  e.  ( 2nd `  x )  /\  v  e.  ( 2nd `  w
)  /\  z  =  ( f  +Q  v
) ) } >. )
69 addclnq 7295 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  v  e.  Q. )  ->  ( f  +Q  v
)  e.  Q. )
7068, 69genpprecll 7434 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( ( z  e.  ( 1st `  A
)  /\  s  e.  ( 1st `  C ) )  ->  ( z  +Q  s )  e.  ( 1st `  ( A  +P.  C ) ) ) )
7164, 67, 70syl2anc 409 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  ( (
z  e.  ( 1st `  A )  /\  s  e.  ( 1st `  C
) )  ->  (
z  +Q  s )  e.  ( 1st `  ( A  +P.  C ) ) ) )
7229, 62, 71mp2and 430 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  ( z  +Q  s )  e.  ( 1st `  ( A  +P.  C ) ) )
7328, 72eqeltrrd 2235 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  w ) )  ->  w  e.  ( 1st `  ( A  +P.  C ) ) )
7427, 73rexlimddv 2579 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  <Q  w )  ->  w  e.  ( 1st `  ( A  +P.  C ) ) )
7574ex 114 . . . . . . . . 9  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  (
z  <Q  w  ->  w  e.  ( 1st `  ( A  +P.  C ) ) ) )
7614ad2antrr 480 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  =  w )  ->  A  <P  B )
77 simpr 109 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  =  w )  ->  z  =  w )
7816adantr 274 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  =  w )  ->  z  e.  ( 1st `  A
) )
7977, 78eqeltrrd 2235 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  =  w )  ->  w  e.  ( 1st `  A
) )
80 ltaddpr 7517 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  A  <P  ( A  +P.  C ) )
818, 65, 80syl2anc 409 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  A  <P  ( A  +P.  C ) )
82 ltprordil 7509 . . . . . . . . . . . . 13  |-  ( A 
<P  ( A  +P.  C
)  ->  ( 1st `  A )  C_  ( 1st `  ( A  +P.  C ) ) )
8382sseld 3127 . . . . . . . . . . . 12  |-  ( A 
<P  ( A  +P.  C
)  ->  ( w  e.  ( 1st `  A
)  ->  w  e.  ( 1st `  ( A  +P.  C ) ) ) )
8481, 83syl 14 . . . . . . . . . . 11  |-  ( A 
<P  B  ->  ( w  e.  ( 1st `  A
)  ->  w  e.  ( 1st `  ( A  +P.  C ) ) ) )
8576, 79, 84sylc 62 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  z  =  w )  ->  w  e.  ( 1st `  ( A  +P.  C ) ) )
8685ex 114 . . . . . . . . 9  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  (
z  =  w  ->  w  e.  ( 1st `  ( A  +P.  C
) ) ) )
87 prcdnql 7404 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( w  <Q  z  ->  w  e.  ( 1st `  A ) ) )
8810, 87sylan 281 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
( w  <Q  z  ->  w  e.  ( 1st `  A ) ) )
8915, 16, 88syl2anc 409 . . . . . . . . . 10  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  (
w  <Q  z  ->  w  e.  ( 1st `  A
) ) )
9015, 89, 84sylsyld 58 . . . . . . . . 9  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  (
w  <Q  z  ->  w  e.  ( 1st `  ( A  +P.  C ) ) ) )
9175, 86, 903jaod 1286 . . . . . . . 8  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  (
( z  <Q  w  \/  z  =  w  \/  w  <Q  z )  ->  w  e.  ( 1st `  ( A  +P.  C ) ) ) )
9224, 91mpd 13 . . . . . . 7  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 1st `  B
) )  /\  (
v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  w  e.  ( 1st `  ( A  +P.  C ) ) )
9392ex 114 . . . . . 6  |-  ( ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  (
w  +Q  v )  e.  ( 1st `  B
) ) )  /\  ( z  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A ) ) )  ->  (
u  <Q  ( z  +Q  v )  ->  w  e.  ( 1st `  ( A  +P.  C ) ) ) )
9493rexlimdvva 2582 . . . . 5  |-  ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  ->  ( E. z  e.  ( 1st `  A
) E. u  e.  ( 2nd `  A
) u  <Q  (
z  +Q  v )  ->  w  e.  ( 1st `  ( A  +P.  C ) ) ) )
9513, 94mpd 13 . . . 4  |-  ( ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  /\  ( v  e.  Q.  /\  ( w  +Q  v
)  e.  ( 1st `  B ) ) )  ->  w  e.  ( 1st `  ( A  +P.  C ) ) )
967, 95rexlimddv 2579 . . 3  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  ->  w  e.  ( 1st `  ( A  +P.  C
) ) )
9796ex 114 . 2  |-  ( A 
<P  B  ->  ( w  e.  ( 1st `  B
)  ->  w  e.  ( 1st `  ( A  +P.  C ) ) ) )
9897ssrdv 3134 1  |-  ( A 
<P  B  ->  ( 1st `  B )  C_  ( 1st `  ( A  +P.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 962    /\ w3a 963    = wceq 1335   E.wex 1472    e. wcel 2128   E.wrex 2436   {crab 2439    C_ wss 3102   <.cop 3563   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   1stc1st 6086   2ndc2nd 6087   Q.cnq 7200    +Q cplq 7202    <Q cltq 7205   P.cnp 7211    +P. cpp 7213    <P cltp 7215
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-iplp 7388  df-iltp 7390
This theorem is referenced by:  ltexpri  7533
  Copyright terms: Public domain W3C validator