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Theorem ltexprlemru 7586
Description: Lemma for ltexpri 7587. 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 7479 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
21brel 4672 . . . . . . 7  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simprd 114 . . . . . 6  |-  ( A 
<P  B  ->  B  e. 
P. )
4 prop 7449 . . . . . 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 7463 . . . . 5  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  w  e.  ( 2nd `  B ) )  ->  E. t  e.  ( 2nd `  B ) t 
<Q  w )
75, 6sylan 283 . . . 4  |-  ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  ->  E. t  e.  ( 2nd `  B ) t 
<Q  w )
8 simprr 531 . . . . . 6  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  ->  t  <Q  w )
9 elprnqu 7456 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  t  e.  ( 2nd `  B ) )  -> 
t  e.  Q. )
105, 9sylan 283 . . . . . . . 8  |-  ( ( A  <P  B  /\  t  e.  ( 2nd `  B ) )  -> 
t  e.  Q. )
1110ad2ant2r 509 . . . . . . 7  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  ->  t  e.  Q. )
12 elprnqu 7456 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  w  e.  ( 2nd `  B ) )  ->  w  e.  Q. )
135, 12sylan 283 . . . . . . . 8  |-  ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  ->  w  e.  Q. )
1413adantr 276 . . . . . . 7  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  ->  w  e.  Q. )
15 ltexnqq 7382 . . . . . . 7  |-  ( ( t  e.  Q.  /\  w  e.  Q. )  ->  ( t  <Q  w  <->  E. v  e.  Q.  (
t  +Q  v )  =  w ) )
1611, 14, 15syl2anc 411 . . . . . 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 147 . . . . 5  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  ->  E. v  e.  Q.  ( t  +Q  v )  =  w )
182simpld 112 . . . . . . . . . 10  |-  ( A 
<P  B  ->  A  e. 
P. )
19 prop 7449 . . . . . . . . . 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 7477 . . . . . . . . 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 283 . . . . . . . 8  |-  ( ( A  <P  B  /\  v  e.  Q. )  ->  E. z  e.  ( 1st `  A ) E. u  e.  ( 2nd `  A ) u  <Q  ( z  +Q  v ) )
2322adantlr 477 . . . . . . 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 509 . . . . . 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 533 . . . . . . . . . . . . 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 488 . . . . . . . . . . . 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 7480 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  B ) ) ) )
2827biimpd 144 . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . 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 539 . . . . . . . . . . . . 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 7461 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  ->  z  <Q  q )
3720, 36syl3an1 1271 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) )  ->  z  <Q  q )
3832, 34, 35, 37syl3anc 1238 . . . . . . . . . . . 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 540 . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 7461 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  q  e.  ( 1st `  B )  /\  t  e.  ( 2nd `  B
) )  ->  q  <Q  t )
445, 43syl3an1 1271 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  q  e.  ( 1st `  B )  /\  t  e.  ( 2nd `  B
) )  ->  q  <Q  t )
4532, 39, 42, 44syl3anc 1238 . . . . . . . . . . . 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 7372 . . . . . . . . . . . . 13  |-  <Q  Or  Q.
47 ltrelnq 7339 . . . . . . . . . . . . 13  |-  <Q  C_  ( Q.  X.  Q. )
4846, 47sotri 5016 . . . . . . . . . . . 12  |-  ( ( z  <Q  q  /\  q  <Q  t )  -> 
z  <Q  t )
4938, 45, 48syl2anc 411 . . . . . . . . . . 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 2597 . . . . . . . . . 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 7383 . . . . . . . . . 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 536 . . . . . . . . . . . 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 488 . . . . . . . . . . 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 531 . . . . . . . . . . . 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 5872 . . . . . . . . . . . . 13  |-  ( ( z  +Q  s )  =  t  ->  (
( z  +Q  s
)  +Q  v )  =  ( t  +Q  v ) )
5756eqeq1d 2184 . . . . . . . . . . . 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 167 . . . . . . . . . 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 7455 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
6120, 60sylan 283 . . . . . . . . . . . . . . . 16  |-  ( ( A  <P  B  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
6261adantlr 477 . . . . . . . . . . . . . . 15  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  z  e.  ( 1st `  A ) )  -> 
z  e.  Q. )
6362ad2ant2r 509 . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . 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 488 . . . . . . . . . . . 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 535 . . . . . . . . . . . . 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 488 . . . . . . . . . . . 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 529 . . . . . . . . . . . 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 7355 . . . . . . . . . . . . 13  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
7069adantl 277 . . . . . . . . . . . 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 7356 . . . . . . . . . . . . 13  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
7271adantl 277 . . . . . . . . . . . 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 6045 . . . . . . . . . . 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 110 . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . 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 7459 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 2nd `  A ) )  -> 
( u  <Q  (
z  +Q  v )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
7720, 76sylan 283 . . . . . . . . . . . . . . 15  |-  ( ( A  <P  B  /\  u  e.  ( 2nd `  A ) )  -> 
( u  <Q  (
z  +Q  v )  ->  ( z  +Q  v )  e.  ( 2nd `  A ) ) )
7826, 75, 77syl2anc 411 . . . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . 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 2252 . . . . . . . . . . . . . 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 2238 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  (
y  e.  ( 1st `  A )  <->  z  e.  ( 1st `  A ) ) )
85 oveq1 5872 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
y  +Q  s )  =  ( z  +Q  s ) )
8685eleq1d 2244 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  (
( y  +Q  s
)  e.  ( 2nd `  B )  <->  ( z  +Q  s )  e.  ( 2nd `  B ) ) )
8784, 86anbi12d 473 . . . . . . . . . . . . . . . 16  |-  ( y  =  z  ->  (
( y  e.  ( 1st `  A )  /\  ( y  +Q  s )  e.  ( 2nd `  B ) )  <->  ( z  e.  ( 1st `  A
)  /\  ( z  +Q  s )  e.  ( 2nd `  B ) ) ) )
8887spcegv 2823 . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 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 7573 . . . . . . . . . . . . 13  |-  ( s  e.  ( 2nd `  C
)  <->  ( s  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  s )  e.  ( 2nd `  B ) ) ) )
9368, 90, 92sylanbrc 417 . . . . . . . . . . . 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 488 . . . . . . . . . . . . . 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 7582 . . . . . . . . . . . . . 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 7442 . . . . . . . . . . . . . 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 7349 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  v  e.  Q. )  ->  ( f  +Q  v
)  e.  Q. )
10098, 99genppreclu 7489 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 433 . . . . . . . . . . 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 2253 . . . . . . . . . 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 2253 . . . . . . . . 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 2597 . . . . . . . 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 115 . . . . . . 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 2600 . . . . . 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 2597 . . . 4  |-  ( ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  /\  ( t  e.  ( 2nd `  B )  /\  t  <Q  w
) )  ->  w  e.  ( 2nd `  ( A  +P.  C ) ) )
1107, 109rexlimddv 2597 . . 3  |-  ( ( A  <P  B  /\  w  e.  ( 2nd `  B ) )  ->  w  e.  ( 2nd `  ( A  +P.  C
) ) )
111110ex 115 . 2  |-  ( A 
<P  B  ->  ( w  e.  ( 2nd `  B
)  ->  w  e.  ( 2nd `  ( A  +P.  C ) ) ) )
112111ssrdv 3159 1  |-  ( A 
<P  B  ->  ( 2nd `  B )  C_  ( 2nd `  ( A  +P.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353   E.wex 1490    e. wcel 2146   E.wrex 2454   {crab 2457    C_ wss 3127   <.cop 3592   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   1stc1st 6129   2ndc2nd 6130   Q.cnq 7254    +Q cplq 7256    <Q cltq 7259   P.cnp 7265    +P. cpp 7267    <P cltp 7269
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-2o 6408  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-enq0 7398  df-nq0 7399  df-0nq0 7400  df-plq0 7401  df-mq0 7402  df-inp 7440  df-iplp 7442  df-iltp 7444
This theorem is referenced by:  ltexpri  7587
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