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Theorem ltexprlemru 7150
Description: Lemma for ltexpri 7151. 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 7043 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
21brel 4478 . . . . . . 7  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simprd 112 . . . . . 6  |-  ( A 
<P  B  ->  B  e. 
P. )
4 prop 7013 . . . . . 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 7027 . . . . 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 7020 . . . . . . . . 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 7020 . . . . . . . . 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 6946 . . . . . . 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 7013 . . . . . . . . . 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 7041 . . . . . . . . 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 7044 . . . . . . . . . . . . . 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 7025 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  ->  z  <Q  q )
3720, 36syl3an1 1207 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  ->  z  <Q  q )
3832, 34, 35, 37syl3anc 1174 . . . . . . . . . . . 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 7025 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  q  e.  ( 1st `  B )  /\  t  e.  ( 2nd `  B
) )  ->  q  <Q  t )
445, 43syl3an1 1207 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  q  e.  ( 1st `  B )  /\  t  e.  ( 2nd `  B
) )  ->  q  <Q  t )
4532, 39, 42, 44syl3anc 1174 . . . . . . . . . . . 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 6936 . . . . . . . . . . . . 13  |-  <Q  Or  Q.
47 ltrelnq 6903 . . . . . . . . . . . . 13  |-  <Q  C_  ( Q.  X.  Q. )
4846, 47sotri 4814 . . . . . . . . . . . 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 2493 . . . . . . . . . 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 6947 . . . . . . . . . 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 5641 . . . . . . . . . . . . 13  |-  ( ( z  +Q  s )  =  t  ->  (
( z  +Q  s
)  +Q  v )  =  ( t  +Q  v ) )
5756eqeq1d 2096 . . . . . . . . . . . 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 7019 . . . . . . . . . . . . . . . . 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 6919 . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . 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 5807 . . . . . . . . . . 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 7023 . . . . . . . . . . . . . . . 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 2164 . . . . . . . . . . . . . 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 2150 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  (
y  e.  ( 1st `  A )  <->  z  e.  ( 1st `  A ) ) )
85 oveq1 5641 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
y  +Q  s )  =  ( z  +Q  s ) )
8685eleq1d 2156 . . . . . . . . . . . . . . . . 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 2707 . . . . . . . . . . . . . . 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 546 . . . . . . . . . . . . . 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 7137 . . . . . . . . . . . . 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 7146 . . . . . . . . . . . . . 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 7006 . . . . . . . . . . . . . 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 6913 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  v  e.  Q. )  ->  ( f  +Q  v
)  e.  Q. )
10098, 99genppreclu 7053 . . . . . . . . . . . . 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 2165 . . . . . . . . . 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 2165 . . . . . . . . 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 2493 . . . . . . . 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 2496 . . . . . 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 2493 . . . 4  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  ->  w  e.  ( 2nd `  ( A  +P.  C ) ) )
1107, 109rexlimddv 2493 . . 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 3029 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 924    = wceq 1289   E.wex 1426    e. wcel 1438   E.wrex 2360   {crab 2363    C_ wss 2997   <.cop 3444   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   1stc1st 5891   2ndc2nd 5892   Q.cnq 6818    +Q cplq 6820    <Q cltq 6823   P.cnp 6829    +P. cpp 6831    <P cltp 6833
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 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
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 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-iltp 7008
This theorem is referenced by:  ltexpri  7151
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