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Theorem addnqprlemrl 7116
Description: Lemma for addnqpr 7120. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
Assertion
Ref Expression
addnqprlemrl  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  C_  ( 1st ` 
<. { l  |  l 
<Q  ( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. ) )
Distinct variable groups:    A, l, u    B, l, u

Proof of Theorem addnqprlemrl
Dummy variables  f  g  h  r  s  t  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqprlu 7106 . . . . . 6  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
2 nqprlu 7106 . . . . . 6  |-  ( B  e.  Q.  ->  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >.  e.  P. )
3 df-iplp 7027 . . . . . . 7  |-  +P.  =  ( x  e.  P. ,  y  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  x )  /\  h  e.  ( 1st `  y
)  /\  f  =  ( g  +Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  x )  /\  h  e.  ( 2nd `  y
)  /\  f  =  ( g  +Q  h
) ) } >. )
4 addclnq 6934 . . . . . . 7  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
53, 4genpelvl 7071 . . . . . 6  |-  ( (
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  e.  P.  /\ 
<. { l  |  l 
<Q  B } ,  {
u  |  B  <Q  u } >.  e.  P. )  ->  ( r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  <->  E. s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
r  =  ( s  +Q  t ) ) )
61, 2, 5syl2an 283 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  <->  E. s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
r  =  ( s  +Q  t ) ) )
76biimpa 290 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )  ->  E. s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
r  =  ( s  +Q  t ) )
8 vex 2622 . . . . . . . . . . . . 13  |-  s  e. 
_V
9 breq1 3848 . . . . . . . . . . . . 13  |-  ( l  =  s  ->  (
l  <Q  A  <->  s  <Q  A ) )
10 ltnqex 7108 . . . . . . . . . . . . . 14  |-  { l  |  l  <Q  A }  e.  _V
11 gtnqex 7109 . . . . . . . . . . . . . 14  |-  { u  |  A  <Q  u }  e.  _V
1210, 11op1st 5917 . . . . . . . . . . . . 13  |-  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  =  { l  |  l 
<Q  A }
138, 9, 12elab2 2763 . . . . . . . . . . . 12  |-  ( s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  s  <Q  A )
1413biimpi 118 . . . . . . . . . . 11  |-  ( s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  s  <Q  A )
1514ad2antrl 474 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  ->  s  <Q  A )
1615adantr 270 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
s  <Q  A )
17 vex 2622 . . . . . . . . . . . . 13  |-  t  e. 
_V
18 breq1 3848 . . . . . . . . . . . . 13  |-  ( l  =  t  ->  (
l  <Q  B  <->  t  <Q  B ) )
19 ltnqex 7108 . . . . . . . . . . . . . 14  |-  { l  |  l  <Q  B }  e.  _V
20 gtnqex 7109 . . . . . . . . . . . . . 14  |-  { u  |  B  <Q  u }  e.  _V
2119, 20op1st 5917 . . . . . . . . . . . . 13  |-  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  =  { l  |  l 
<Q  B }
2217, 18, 21elab2 2763 . . . . . . . . . . . 12  |-  ( t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  <->  t  <Q  B )
2322biimpi 118 . . . . . . . . . . 11  |-  ( t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  ->  t  <Q  B )
2423ad2antll 475 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  ->  t  <Q  B )
2524adantr 270 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
t  <Q  B )
26 ltrelnq 6924 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
2726brel 4490 . . . . . . . . . . 11  |-  ( s 
<Q  A  ->  ( s  e.  Q.  /\  A  e.  Q. ) )
2816, 27syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( s  e.  Q.  /\  A  e.  Q. )
)
2926brel 4490 . . . . . . . . . . 11  |-  ( t 
<Q  B  ->  ( t  e.  Q.  /\  B  e.  Q. ) )
3025, 29syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( t  e.  Q.  /\  B  e.  Q. )
)
31 lt2addnq 6963 . . . . . . . . . 10  |-  ( ( ( s  e.  Q.  /\  A  e.  Q. )  /\  ( t  e.  Q.  /\  B  e.  Q. )
)  ->  ( (
s  <Q  A  /\  t  <Q  B )  ->  (
s  +Q  t ) 
<Q  ( A  +Q  B
) ) )
3228, 30, 31syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( ( s  <Q  A  /\  t  <Q  B )  ->  ( s  +Q  t )  <Q  ( A  +Q  B ) ) )
3316, 25, 32mp2and 424 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( s  +Q  t
)  <Q  ( A  +Q  B ) )
34 breq1 3848 . . . . . . . . 9  |-  ( r  =  ( s  +Q  t )  ->  (
r  <Q  ( A  +Q  B )  <->  ( s  +Q  t )  <Q  ( A  +Q  B ) ) )
3534adantl 271 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( r  <Q  ( A  +Q  B )  <->  ( s  +Q  t )  <Q  ( A  +Q  B ) ) )
3633, 35mpbird 165 . . . . . . 7  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
r  <Q  ( A  +Q  B ) )
37 vex 2622 . . . . . . . 8  |-  r  e. 
_V
38 breq1 3848 . . . . . . . 8  |-  ( l  =  r  ->  (
l  <Q  ( A  +Q  B )  <->  r  <Q  ( A  +Q  B ) ) )
39 ltnqex 7108 . . . . . . . . 9  |-  { l  |  l  <Q  ( A  +Q  B ) }  e.  _V
40 gtnqex 7109 . . . . . . . . 9  |-  { u  |  ( A  +Q  B )  <Q  u }  e.  _V
4139, 40op1st 5917 . . . . . . . 8  |-  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. )  =  {
l  |  l  <Q 
( A  +Q  B
) }
4237, 38, 41elab2 2763 . . . . . . 7  |-  ( r  e.  ( 1st `  <. { l  |  l  <Q 
( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. )  <->  r  <Q  ( A  +Q  B ) )
4336, 42sylibr 132 . . . . . 6  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
r  e.  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. ) )
4443ex 113 . . . . 5  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  ->  (
r  =  ( s  +Q  t )  -> 
r  e.  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. ) ) )
4544rexlimdvva 2496 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )  -> 
( E. s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) r  =  ( s  +Q  t )  ->  r  e.  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B
)  <Q  u } >. ) ) )
467, 45mpd 13 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 1st `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )  -> 
r  e.  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. ) )
4746ex 113 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( r  e.  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  ->  r  e.  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B
)  <Q  u } >. ) ) )
4847ssrdv 3031 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 1st `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  C_  ( 1st ` 
<. { l  |  l 
<Q  ( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074   E.wrex 2360    C_ wss 2999   <.cop 3449   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   1stc1st 5909   Q.cnq 6839    +Q cplq 6841    <Q cltq 6844   P.cnp 6850    +P. cpp 6852
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-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-inp 7025  df-iplp 7027
This theorem is referenced by:  addnqprlemfu  7119  addnqpr  7120
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