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Theorem ltexprlemrl 7609
Description: Lemma for ltexpri 7612. 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 7504 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
21brel 4679 . . . . . . 7  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simprd 114 . . . . . 6  |-  ( A 
<P  B  ->  B  e. 
P. )
4 prop 7474 . . . . . 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 7489 . . . . 5  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  w  e.  ( 1st `  B ) )  ->  E. v  e.  Q.  ( w  +Q  v
)  e.  ( 1st `  B ) )
75, 6sylan 283 . . . 4  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  ->  E. v  e.  Q.  ( w  +Q  v
)  e.  ( 1st `  B ) )
82simpld 112 . . . . . . . 8  |-  ( A 
<P  B  ->  A  e. 
P. )
9 prop 7474 . . . . . . . 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 7502 . . . . . . 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 283 . . . . . 6  |-  ( ( A  <P  B  /\  v  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( z  +Q  v ) )
1312ad2ant2r 509 . . . . 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 533 . . . . . . . . . . 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 276 . . . . . . . . . 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 535 . . . . . . . . . 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 7480 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
1810, 17sylan 283 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
1915, 16, 18syl2anc 411 . . . . . . . . 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 7480 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  w  e.  ( 1st `  B ) )  ->  w  e.  Q. )
215, 20sylan 283 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  ->  w  e.  Q. )
2221ad3antrrr 492 . . . . . . . . 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 7395 . . . . . . . . 9  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  <Q  w  \/  z  =  w  \/  w  <Q  z ) )
2419, 22, 23syl2anc 411 . . . . . . . 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 7407 . . . . . . . . . . . . 13  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  <Q  w  <->  E. s  e.  Q.  (
z  +Q  s )  =  w ) )
2619, 22, 25syl2anc 411 . . . . . . . . . . . 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 296 . . . . . . . . . . 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 531 . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . 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 110 . . . . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . . . . 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 7484 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 2nd `  A ) )  -> 
( u  <Q  (
z  +Q  v )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
3410, 33sylan 283 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  <P  B  /\  u  e.  ( 2nd `  A ) )  -> 
( u  <Q  (
z  +Q  v )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
3515, 32, 34syl2anc 411 . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . . . 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 7380 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4241adantl 277 . . . . . . . . . . . . . . . . 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 7381 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
4443adantl 277 . . . . . . . . . . . . . . . . 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 6055 . . . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . . . 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 5882 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  +Q  s )  =  w  ->  (
( z  +Q  s
)  +Q  v )  =  ( w  +Q  v ) )
4948eleq1d 2246 . . . . . . . . . . . . . . . . . 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 167 . . . . . . . . . . . . . . . 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 2254 . . . . . . . . . . . . . . 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 2240 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( z  +Q  v )  ->  (
y  e.  ( 2nd `  A )  <->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
54 oveq1 5882 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( z  +Q  v )  ->  (
y  +Q  s )  =  ( ( z  +Q  v )  +Q  s ) )
5554eleq1d 2246 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( z  +Q  v )  ->  (
( y  +Q  s
)  e.  ( 1st `  B )  <->  ( (
z  +Q  v )  +Q  s )  e.  ( 1st `  B
) ) )
5653, 55anbi12d 473 . . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . 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 7597 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 1st `  C
)  <->  ( s  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  s )  e.  ( 1st `  B ) ) ) )
6230, 59, 61sylanbrc 417 . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . 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 7607 . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . 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 7467 . . . . . . . . . . . . . . 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 7374 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  v  e.  Q. )  ->  ( f  +Q  v
)  e.  Q. )
7068, 69genpprecll 7513 . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 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 433 . . . . . . . . . . . 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 2255 . . . . . . . . . . 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 2599 . . . . . . . . . 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 115 . . . . . . . . 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 488 . . . . . . . . . . 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 110 . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 2255 . . . . . . . . . . 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 7596 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  A  <P  ( A  +P.  C ) )
818, 65, 80syl2anc 411 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  A  <P  ( A  +P.  C ) )
82 ltprordil 7588 . . . . . . . . . . . . 13  |-  ( A 
<P  ( A  +P.  C
)  ->  ( 1st `  A )  C_  ( 1st `  ( A  +P.  C ) ) )
8382sseld 3155 . . . . . . . . . . . 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 115 . . . . . . . . 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 7483 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( w  <Q  z  ->  w  e.  ( 1st `  A ) ) )
8810, 87sylan 283 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
( w  <Q  z  ->  w  e.  ( 1st `  A ) ) )
8915, 16, 88syl2anc 411 . . . . . . . . . 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 1304 . . . . . . . 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 115 . . . . . 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 2602 . . . . 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 2599 . . 3  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  ->  w  e.  ( 1st `  ( A  +P.  C
) ) )
9796ex 115 . 2  |-  ( A 
<P  B  ->  ( w  e.  ( 1st `  B
)  ->  w  e.  ( 1st `  ( A  +P.  C ) ) ) )
9897ssrdv 3162 1  |-  ( A 
<P  B  ->  ( 1st `  B )  C_  ( 1st `  ( A  +P.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ w3o 977    /\ w3a 978    = wceq 1353   E.wex 1492    e. wcel 2148   E.wrex 2456   {crab 2459    C_ wss 3130   <.cop 3596   class class class wbr 4004   ` cfv 5217  (class class class)co 5875   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279    +Q cplq 7281    <Q cltq 7284   P.cnp 7290    +P. cpp 7292    <P cltp 7294
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 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-iplp 7467  df-iltp 7469
This theorem is referenced by:  ltexpri  7612
  Copyright terms: Public domain W3C validator