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Theorem ltexprlemrl 7169
Description: Lemma for ltexpri 7172. 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 7064 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
21brel 4490 . . . . . . 7  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simprd 112 . . . . . 6  |-  ( A 
<P  B  ->  B  e. 
P. )
4 prop 7034 . . . . . 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 7049 . . . . 5  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  w  e.  ( 1st `  B ) )  ->  E. v  e.  Q.  ( w  +Q  v
)  e.  ( 1st `  B ) )
75, 6sylan 277 . . . 4  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  ->  E. v  e.  Q.  ( w  +Q  v
)  e.  ( 1st `  B ) )
82simpld 110 . . . . . . . 8  |-  ( A 
<P  B  ->  A  e. 
P. )
9 prop 7034 . . . . . . . 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 7062 . . . . . . 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 277 . . . . . 6  |-  ( ( A  <P  B  /\  v  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( z  +Q  v ) )
1312ad2ant2r 493 . . . . 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 500 . . . . . . . . . . 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 270 . . . . . . . . . 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 502 . . . . . . . . . 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 7040 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
1810, 17sylan 277 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
1915, 16, 18syl2anc 403 . . . . . . . . 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 7040 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  w  e.  ( 1st `  B ) )  ->  w  e.  Q. )
215, 20sylan 277 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  ->  w  e.  Q. )
2221ad3antrrr 476 . . . . . . . . 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 6955 . . . . . . . . 9  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  <Q  w  \/  z  =  w  \/  w  <Q  z ) )
2419, 22, 23syl2anc 403 . . . . . . . 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 6967 . . . . . . . . . . . . 13  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  <Q  w  <->  E. s  e.  Q.  (
z  +Q  s )  =  w ) )
2619, 22, 25syl2anc 403 . . . . . . . . . . . 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 290 . . . . . . . . . . 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 499 . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 498 . . . . . . . . . . . . . 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 108 . . . . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . . . . 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 7044 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 2nd `  A ) )  -> 
( u  <Q  (
z  +Q  v )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
3410, 33sylan 277 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  <P  B  /\  u  e.  ( 2nd `  A ) )  -> 
( u  <Q  (
z  +Q  v )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
3515, 32, 34syl2anc 403 . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . 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 502 . . . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . 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 6940 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4241adantl 271 . . . . . . . . . . . . . . . . 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 6941 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
4443adantl 271 . . . . . . . . . . . . . . . . 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 5825 . . . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . 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 5659 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  +Q  s )  =  w  ->  (
( z  +Q  s
)  +Q  v )  =  ( w  +Q  v ) )
4948eleq1d 2156 . . . . . . . . . . . . . . . . . 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 165 . . . . . . . . . . . . . . . 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 2164 . . . . . . . . . . . . . . 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 2150 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( z  +Q  v )  ->  (
y  e.  ( 2nd `  A )  <->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
54 oveq1 5659 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( z  +Q  v )  ->  (
y  +Q  s )  =  ( ( z  +Q  v )  +Q  s ) )
5554eleq1d 2156 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( z  +Q  v )  ->  (
( y  +Q  s
)  e.  ( 1st `  B )  <->  ( (
z  +Q  v )  +Q  s )  e.  ( 1st `  B
) ) )
5653, 55anbi12d 457 . . . . . . . . . . . . . . . . 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 2707 . . . . . . . . . . . . . . . 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 546 . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . 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 7157 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 1st `  C
)  <->  ( s  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  s )  e.  ( 1st `  B ) ) ) )
6230, 59, 61sylanbrc 408 . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 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 7167 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 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 7027 . . . . . . . . . . . . . . 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 6934 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  v  e.  Q. )  ->  ( f  +Q  v
)  e.  Q. )
7068, 69genpprecll 7073 . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 2165 . . . . . . . . . . 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 2493 . . . . . . . . . 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 113 . . . . . . . . 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 472 . . . . . . . . . . 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 108 . . . . . . . . . . . 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 270 . . . . . . . . . . . 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 2165 . . . . . . . . . . 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 7156 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  A  <P  ( A  +P.  C ) )
818, 65, 80syl2anc 403 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  A  <P  ( A  +P.  C ) )
82 ltprordil 7148 . . . . . . . . . . . . 13  |-  ( A 
<P  ( A  +P.  C
)  ->  ( 1st `  A )  C_  ( 1st `  ( A  +P.  C ) ) )
8382sseld 3024 . . . . . . . . . . . 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 61 . . . . . . . . . 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 113 . . . . . . . . 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 7043 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
( w  <Q  z  ->  w  e.  ( 1st `  A ) ) )
8810, 87sylan 277 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
( w  <Q  z  ->  w  e.  ( 1st `  A ) ) )
8915, 16, 88syl2anc 403 . . . . . . . . . 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 57 . . . . . . . . 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 1240 . . . . . . . 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 113 . . . . . 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 2496 . . . . 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 2493 . . 3  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  B ) )  ->  w  e.  ( 1st `  ( A  +P.  C
) ) )
9796ex 113 . 2  |-  ( A 
<P  B  ->  ( w  e.  ( 1st `  B
)  ->  w  e.  ( 1st `  ( A  +P.  C ) ) ) )
9897ssrdv 3031 1  |-  ( A 
<P  B  ->  ( 1st `  B )  C_  ( 1st `  ( A  +P.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ w3o 923    /\ w3a 924    = wceq 1289   E.wex 1426    e. wcel 1438   E.wrex 2360   {crab 2363    C_ wss 2999   <.cop 3449   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   1stc1st 5909   2ndc2nd 5910   Q.cnq 6839    +Q cplq 6841    <Q cltq 6844   P.cnp 6850    +P. cpp 6852    <P cltp 6854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-enq0 6983  df-nq0 6984  df-0nq0 6985  df-plq0 6986  df-mq0 6987  df-inp 7025  df-iplp 7027  df-iltp 7029
This theorem is referenced by:  ltexpri  7172
  Copyright terms: Public domain W3C validator