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Theorem ltexprlemrl 6766
Description: Lemma for ltexpri 6769. Reverse directon 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 6661 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
21brel 4420 . . . . . . 7  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simprd 111 . . . . . 6  |-  ( A 
<P  B  ->  B  e. 
P. )
4 prop 6631 . . . . . 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 6646 . . . . 5  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  w  e.  ( 1st `  B ) )  ->  E. v  e.  Q.  ( w  +Q  v
)  e.  ( 1st `  B ) )
75, 6sylan 271 . . . 4  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  ->  E. v  e.  Q.  ( w  +Q  v
)  e.  ( 1st `  B ) )
82simpld 109 . . . . . . . 8  |-  ( A 
<P  B  ->  A  e. 
P. )
9 prop 6631 . . . . . . . 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 6659 . . . . . . 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 271 . . . . . 6  |-  ( ( A  <P  B  /\  v  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( z  +Q  v ) )
1312ad2ant2r 486 . . . . 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 493 . . . . . . . . . . 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 265 . . . . . . . . . 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 495 . . . . . . . . . 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 6637 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
1810, 17sylan 271 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
1915, 16, 18syl2anc 397 . . . . . . . . 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 6637 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  w  e.  ( 1st `  B ) )  ->  w  e.  Q. )
215, 20sylan 271 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  ->  w  e.  Q. )
2221ad3antrrr 469 . . . . . . . . 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 6552 . . . . . . . . 9  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  <Q  w  \/  z  =  w  \/  w  <Q  z ) )
2419, 22, 23syl2anc 397 . . . . . . . 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 6564 . . . . . . . . . . . . 13  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  <Q  w  <->  E. s  e.  Q.  (
z  +Q  s )  =  w ) )
2619, 22, 25syl2anc 397 . . . . . . . . . . . 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 284 . . . . . . . . . . 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 492 . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . 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 107 . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . . . 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 6641 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 2nd `  A ) )  -> 
( u  <Q  (
z  +Q  v )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
3410, 33sylan 271 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  <P  B  /\  u  e.  ( 2nd `  A ) )  -> 
( u  <Q  (
z  +Q  v )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
3515, 32, 34syl2anc 397 . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . . . . 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 6537 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4241adantl 266 . . . . . . . . . . . . . . . . 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 6538 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
4443adantl 266 . . . . . . . . . . . . . . . . 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 5709 . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . . . . 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 5547 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  +Q  s )  =  w  ->  (
( z  +Q  s
)  +Q  v )  =  ( w  +Q  v ) )
4948eleq1d 2122 . . . . . . . . . . . . . . . . . 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 160 . . . . . . . . . . . . . . . 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 2130 . . . . . . . . . . . . . . 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 2116 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( z  +Q  v )  ->  (
y  e.  ( 2nd `  A )  <->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
54 oveq1 5547 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( z  +Q  v )  ->  (
y  +Q  s )  =  ( ( z  +Q  v )  +Q  s ) )
5554eleq1d 2122 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( z  +Q  v )  ->  (
( y  +Q  s
)  e.  ( 1st `  B )  <->  ( (
z  +Q  v )  +Q  s )  e.  ( 1st `  B
) ) )
5653, 55anbi12d 450 . . . . . . . . . . . . . . . . 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 2658 . . . . . . . . . . . . . . . 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 521 . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . 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 6754 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 1st `  C
)  <->  ( s  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  s )  e.  ( 1st `  B ) ) ) )
6230, 59, 61sylanbrc 402 . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 6764 . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 6624 . . . . . . . . . . . . . . 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 6531 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  v  e.  Q. )  ->  ( f  +Q  v
)  e.  Q. )
7068, 69genpprecll 6670 . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . 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 417 . . . . . . . . . . . 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 2131 . . . . . . . . . . 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 2454 . . . . . . . . . 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 112 . . . . . . . . 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 465 . . . . . . . . . . 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 107 . . . . . . . . . . . 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 265 . . . . . . . . . . . 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 2131 . . . . . . . . . . 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 6753 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  A  <P  ( A  +P.  C ) )
818, 65, 80syl2anc 397 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  A  <P  ( A  +P.  C ) )
82 ltprordil 6745 . . . . . . . . . . . . 13  |-  ( A 
<P  ( A  +P.  C
)  ->  ( 1st `  A )  C_  ( 1st `  ( A  +P.  C ) ) )
8382sseld 2972 . . . . . . . . . . . 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 60 . . . . . . . . . 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 112 . . . . . . . . 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 6640 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( w  <Q  z  ->  w  e.  ( 1st `  A ) ) )
8810, 87sylan 271 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
( w  <Q  z  ->  w  e.  ( 1st `  A ) ) )
8915, 16, 88syl2anc 397 . . . . . . . . . 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 56 . . . . . . . . 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 1210 . . . . . . . 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 112 . . . . . 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 2457 . . . . 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 2454 . . 3  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  ->  w  e.  ( 1st `  ( A  +P.  C
) ) )
9796ex 112 . 2  |-  ( A 
<P  B  ->  ( w  e.  ( 1st `  B
)  ->  w  e.  ( 1st `  ( A  +P.  C ) ) ) )
9897ssrdv 2979 1  |-  ( A 
<P  B  ->  ( 1st `  B )  C_  ( 1st `  ( A  +P.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    \/ w3o 895    /\ w3a 896    = wceq 1259   E.wex 1397    e. wcel 1409   E.wrex 2324   {crab 2327    C_ wss 2945   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    +Q cplq 6438    <Q cltq 6441   P.cnp 6447    +P. cpp 6449    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624  df-iltp 6626
This theorem is referenced by:  ltexpri  6769
  Copyright terms: Public domain W3C validator