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Theorem ltexprlemru 6864
Description: Lemma for ltexpri 6865. One direction of our result for upper 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
ltexprlemru  |-  ( A 
<P  B  ->  ( 2nd `  B )  C_  ( 2nd `  ( A  +P.  C ) ) )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y

Proof of Theorem ltexprlemru
Dummy variables  z  w  u  v  f  g  h  q  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 6757 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
21brel 4418 . . . . . . 7  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simprd 112 . . . . . 6  |-  ( A 
<P  B  ->  B  e. 
P. )
4 prop 6727 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
53, 4syl 14 . . . . 5  |-  ( A 
<P  B  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
6 prnminu 6741 . . . . 5  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  w  e.  ( 2nd `  B ) )  ->  E. t  e.  ( 2nd `  B ) t 
<Q  w )
75, 6sylan 277 . . . 4  |-  ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  ->  E. t  e.  ( 2nd `  B ) t 
<Q  w )
8 simprr 499 . . . . . 6  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  ->  t  <Q  w )
9 elprnqu 6734 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  t  e.  ( 2nd `  B ) )  -> 
t  e.  Q. )
105, 9sylan 277 . . . . . . . 8  |-  ( ( A  <P  B  /\  t  e.  ( 2nd `  B ) )  -> 
t  e.  Q. )
1110ad2ant2r 493 . . . . . . 7  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  ->  t  e.  Q. )
12 elprnqu 6734 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  w  e.  ( 2nd `  B ) )  ->  w  e.  Q. )
135, 12sylan 277 . . . . . . . 8  |-  ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  ->  w  e.  Q. )
1413adantr 270 . . . . . . 7  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  ->  w  e.  Q. )
15 ltexnqq 6660 . . . . . . 7  |-  ( ( t  e.  Q.  /\  w  e.  Q. )  ->  ( t  <Q  w  <->  E. v  e.  Q.  (
t  +Q  v )  =  w ) )
1611, 14, 15syl2anc 403 . . . . . 6  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  ->  (
t  <Q  w  <->  E. v  e.  Q.  ( t  +Q  v )  =  w ) )
178, 16mpbid 145 . . . . 5  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  ->  E. v  e.  Q.  ( t  +Q  v )  =  w )
182simpld 110 . . . . . . . . . 10  |-  ( A 
<P  B  ->  A  e. 
P. )
19 prop 6727 . . . . . . . . . 10  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2018, 19syl 14 . . . . . . . . 9  |-  ( A 
<P  B  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
21 prarloc 6755 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  v  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( z  +Q  v ) )
2220, 21sylan 277 . . . . . . . 8  |-  ( ( A  <P  B  /\  v  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( z  +Q  v ) )
2322adantlr 461 . . . . . . 7  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  v  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( z  +Q  v ) )
2423ad2ant2r 493 . . . . . 6  |-  ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B
)  /\  t  <Q  w ) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  ->  E. z  e.  ( 1st `  A
) E. u  e.  ( 2nd `  A
) u  <Q  (
z  +Q  v ) )
25 simplll 500 . . . . . . . . . . . . 13  |-  ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B
)  /\  t  <Q  w ) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  ->  A  <P  B )
2625ad2antrr 472 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  A  <P  B )
27 ltdfpr 6758 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  B ) ) ) )
2827biimpd 142 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  ->  E. q  e.  Q.  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  B ) ) ) )
292, 28mpcom 36 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  E. q  e.  Q.  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  B ) ) )
3026, 29syl 14 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  E. q  e.  Q.  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  B ) ) )
3125adantr 270 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 2nd `  B
) )  /\  (
t  e.  ( 2nd `  B )  /\  t  <Q  w ) )  /\  ( v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  ->  A  <P  B )
3231ad2antrr 472 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
q  e.  Q.  /\  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  B ) ) ) )  ->  A  <P  B )
33 simplrl 502 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  z  e.  ( 1st `  A
) )
3433adantr 270 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
q  e.  Q.  /\  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  B ) ) ) )  -> 
z  e.  ( 1st `  A ) )
35 simprrl 506 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
q  e.  Q.  /\  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  B ) ) ) )  -> 
q  e.  ( 2nd `  A ) )
36 prltlu 6739 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  ->  z  <Q  q )
3720, 36syl3an1 1203 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  ->  z  <Q  q )
3832, 34, 35, 37syl3anc 1170 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
q  e.  Q.  /\  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  B ) ) ) )  -> 
z  <Q  q )
39 simprrr 507 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
q  e.  Q.  /\  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  B ) ) ) )  -> 
q  e.  ( 1st `  B ) )
40 simplrl 502 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B
)  /\  t  <Q  w ) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  ->  t  e.  ( 2nd `  B ) )
4140adantr 270 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 2nd `  B
) )  /\  (
t  e.  ( 2nd `  B )  /\  t  <Q  w ) )  /\  ( v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  ->  t  e.  ( 2nd `  B
) )
4241ad2antrr 472 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
q  e.  Q.  /\  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  B ) ) ) )  -> 
t  e.  ( 2nd `  B ) )
43 prltlu 6739 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  q  e.  ( 1st `  B )  /\  t  e.  ( 2nd `  B
) )  ->  q  <Q  t )
445, 43syl3an1 1203 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  q  e.  ( 1st `  B )  /\  t  e.  ( 2nd `  B
) )  ->  q  <Q  t )
4532, 39, 42, 44syl3anc 1170 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
q  e.  Q.  /\  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  B ) ) ) )  -> 
q  <Q  t )
46 ltsonq 6650 . . . . . . . . . . . . 13  |-  <Q  Or  Q.
47 ltrelnq 6617 . . . . . . . . . . . . 13  |-  <Q  C_  ( Q.  X.  Q. )
4846, 47sotri 4750 . . . . . . . . . . . 12  |-  ( ( z  <Q  q  /\  q  <Q  t )  -> 
z  <Q  t )
4938, 45, 48syl2anc 403 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
q  e.  Q.  /\  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  B ) ) ) )  -> 
z  <Q  t )
5030, 49rexlimddv 2482 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  z  <Q  t )
51 ltexnqi 6661 . . . . . . . . . 10  |-  ( z 
<Q  t  ->  E. s  e.  Q.  ( z  +Q  s )  =  t )
5250, 51syl 14 . . . . . . . . 9  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  E. s  e.  Q.  ( z  +Q  s )  =  t )
53 simplrr 503 . . . . . . . . . . . 12  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 2nd `  B
) )  /\  (
t  e.  ( 2nd `  B )  /\  t  <Q  w ) )  /\  ( v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  ->  (
t  +Q  v )  =  w )
5453ad2antrr 472 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  ( t  +Q  v )  =  w )
55 simprr 499 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  ( z  +Q  s )  =  t )
56 oveq1 5550 . . . . . . . . . . . . 13  |-  ( ( z  +Q  s )  =  t  ->  (
( z  +Q  s
)  +Q  v )  =  ( t  +Q  v ) )
5756eqeq1d 2090 . . . . . . . . . . . 12  |-  ( ( z  +Q  s )  =  t  ->  (
( ( z  +Q  s )  +Q  v
)  =  w  <->  ( t  +Q  v )  =  w ) )
5855, 57syl 14 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  ( (
( z  +Q  s
)  +Q  v )  =  w  <->  ( t  +Q  v )  =  w ) )
5954, 58mpbird 165 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  ( (
z  +Q  s )  +Q  v )  =  w )
60 elprnql 6733 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
6120, 60sylan 277 . . . . . . . . . . . . . . . 16  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
6261adantlr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
6362ad2ant2r 493 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B
)  /\  t  <Q  w ) )  /\  (
z  e.  ( 1st `  A )  /\  u  e.  ( 2nd `  A
) ) )  -> 
z  e.  Q. )
6463adantlr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 2nd `  B
) )  /\  (
t  e.  ( 2nd `  B )  /\  t  <Q  w ) )  /\  ( v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  ->  z  e.  Q. )
6564ad2antrr 472 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  z  e.  Q. )
66 simplrl 502 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 2nd `  B
) )  /\  (
t  e.  ( 2nd `  B )  /\  t  <Q  w ) )  /\  ( v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  ->  v  e.  Q. )
6766ad2antrr 472 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  v  e.  Q. )
68 simprl 498 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  s  e.  Q. )
69 addcomnqg 6633 . . . . . . . . . . . . 13  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
7069adantl 271 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B
)  /\  t  <Q  w ) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  /\  ( f  e.  Q.  /\  g  e.  Q. ) )  -> 
( f  +Q  g
)  =  ( g  +Q  f ) )
71 addassnqg 6634 . . . . . . . . . . . . 13  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
7271adantl 271 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B
)  /\  t  <Q  w ) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e. 
Q. ) )  -> 
( ( f  +Q  g )  +Q  h
)  =  ( f  +Q  ( g  +Q  h ) ) )
7365, 67, 68, 70, 72caov32d 5712 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  ( (
z  +Q  v )  +Q  s )  =  ( ( z  +Q  s )  +Q  v
) )
74 simpr 108 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  u  <Q  ( z  +Q  v
) )
75 simplrr 503 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  u  e.  ( 2nd `  A
) )
76 prcunqu 6737 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 2nd `  A ) )  -> 
( u  <Q  (
z  +Q  v )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
7720, 76sylan 277 . . . . . . . . . . . . . . 15  |-  ( ( A  <P  B  /\  u  e.  ( 2nd `  A ) )  -> 
( u  <Q  (
z  +Q  v )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
7826, 75, 77syl2anc 403 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  (
u  <Q  ( z  +Q  v )  ->  (
z  +Q  v )  e.  ( 2nd `  A
) ) )
7974, 78mpd 13 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  (
z  +Q  v )  e.  ( 2nd `  A
) )
8079adantr 270 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) )
8133adantr 270 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  z  e.  ( 1st `  A ) )
8241ad2antrr 472 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  t  e.  ( 2nd `  B ) )
8355, 82eqeltrd 2156 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  ( z  +Q  s )  e.  ( 2nd `  B ) )
84 eleq1 2142 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  (
y  e.  ( 1st `  A )  <->  z  e.  ( 1st `  A ) ) )
85 oveq1 5550 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
y  +Q  s )  =  ( z  +Q  s ) )
8685eleq1d 2148 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  (
( y  +Q  s
)  e.  ( 2nd `  B )  <->  ( z  +Q  s )  e.  ( 2nd `  B ) ) )
8784, 86anbi12d 457 . . . . . . . . . . . . . . . 16  |-  ( y  =  z  ->  (
( y  e.  ( 1st `  A )  /\  ( y  +Q  s )  e.  ( 2nd `  B ) )  <->  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  s )  e.  ( 2nd `  B ) ) ) )
8887spcegv 2687 . . . . . . . . . . . . . . 15  |-  ( z  e.  ( 1st `  A
)  ->  ( (
z  e.  ( 1st `  A )  /\  (
z  +Q  s )  e.  ( 2nd `  B
) )  ->  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  s )  e.  ( 2nd `  B ) ) ) )
8988anabsi5 544 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ( 1st `  A )  /\  (
z  +Q  s )  e.  ( 2nd `  B
) )  ->  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  s )  e.  ( 2nd `  B ) ) )
9081, 83, 89syl2anc 403 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  s )  e.  ( 2nd `  B ) ) )
91 ltexprlem.1 . . . . . . . . . . . . . 14  |-  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 ) ) } >.
9291ltexprlemelu 6851 . . . . . . . . . . . . 13  |-  ( s  e.  ( 2nd `  C
)  <->  ( s  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  s )  e.  ( 2nd `  B ) ) ) )
9368, 90, 92sylanbrc 408 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  s  e.  ( 2nd `  C ) )
9431ad2antrr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  A  <P  B )
9594, 18syl 14 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  A  e.  P. )
9691ltexprlempr 6860 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  C  e. 
P. )
9794, 96syl 14 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  C  e.  P. )
98 df-iplp 6720 . . . . . . . . . . . . . 14  |-  +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
) ) } >. )
99 addclnq 6627 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  v  e.  Q. )  ->  ( f  +Q  v
)  e.  Q. )
10098, 99genppreclu 6767 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( ( ( z  +Q  v )  e.  ( 2nd `  A
)  /\  s  e.  ( 2nd `  C ) )  ->  ( (
z  +Q  v )  +Q  s )  e.  ( 2nd `  ( A  +P.  C ) ) ) )
10195, 97, 100syl2anc 403 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  ( (
( z  +Q  v
)  e.  ( 2nd `  A )  /\  s  e.  ( 2nd `  C
) )  ->  (
( z  +Q  v
)  +Q  s )  e.  ( 2nd `  ( A  +P.  C ) ) ) )
10280, 93, 101mp2and 424 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  ( (
z  +Q  v )  +Q  s )  e.  ( 2nd `  ( A  +P.  C ) ) )
10373, 102eqeltrrd 2157 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  ( (
z  +Q  s )  +Q  v )  e.  ( 2nd `  ( A  +P.  C ) ) )
10459, 103eqeltrrd 2157 . . . . . . . . 9  |-  ( ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  /\  (
s  e.  Q.  /\  ( z  +Q  s
)  =  t ) )  ->  w  e.  ( 2nd `  ( A  +P.  C ) ) )
10552, 104rexlimddv 2482 . . . . . . . 8  |-  ( ( ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  /\  u  <Q  ( z  +Q  v
) )  ->  w  e.  ( 2nd `  ( A  +P.  C ) ) )
106105ex 113 . . . . . . 7  |-  ( ( ( ( ( A 
<P  B  /\  w  e.  ( 2nd `  B
) )  /\  (
t  e.  ( 2nd `  B )  /\  t  <Q  w ) )  /\  ( v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  /\  ( z  e.  ( 1st `  A
)  /\  u  e.  ( 2nd `  A ) ) )  ->  (
u  <Q  ( z  +Q  v )  ->  w  e.  ( 2nd `  ( A  +P.  C ) ) ) )
107106rexlimdvva 2485 . . . . . 6  |-  ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B
)  /\  t  <Q  w ) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  ->  ( E. z  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A
) u  <Q  (
z  +Q  v )  ->  w  e.  ( 2nd `  ( A  +P.  C ) ) ) )
10824, 107mpd 13 . . . . 5  |-  ( ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B
)  /\  t  <Q  w ) )  /\  (
v  e.  Q.  /\  ( t  +Q  v
)  =  w ) )  ->  w  e.  ( 2nd `  ( A  +P.  C ) ) )
10917, 108rexlimddv 2482 . . . 4  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  ->  w  e.  ( 2nd `  ( A  +P.  C ) ) )
1107, 109rexlimddv 2482 . . 3  |-  ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  ->  w  e.  ( 2nd `  ( A  +P.  C
) ) )
111110ex 113 . 2  |-  ( A 
<P  B  ->  ( w  e.  ( 2nd `  B
)  ->  w  e.  ( 2nd `  ( A  +P.  C ) ) ) )
112111ssrdv 3006 1  |-  ( A 
<P  B  ->  ( 2nd `  B )  C_  ( 2nd `  ( A  +P.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285   E.wex 1422    e. wcel 1434   E.wrex 2350   {crab 2353    C_ wss 2974   <.cop 3409   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   1stc1st 5796   2ndc2nd 5797   Q.cnq 6532    +Q cplq 6534    <Q cltq 6537   P.cnp 6543    +P. cpp 6545    <P cltp 6547
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722
This theorem is referenced by:  ltexpri  6865
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