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Theorem addnqprlemrl 7333
Description: Lemma for addnqpr 7337. 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 7323 . . . . . 6  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
2 nqprlu 7323 . . . . . 6  |-  ( B  e.  Q.  ->  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >.  e.  P. )
3 df-iplp 7244 . . . . . . 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 7151 . . . . . . 7  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
53, 4genpelvl 7288 . . . . . 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 287 . . . . 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 294 . . . 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 2663 . . . . . . . . . . . . 13  |-  s  e. 
_V
9 breq1 3902 . . . . . . . . . . . . 13  |-  ( l  =  s  ->  (
l  <Q  A  <->  s  <Q  A ) )
10 ltnqex 7325 . . . . . . . . . . . . . 14  |-  { l  |  l  <Q  A }  e.  _V
11 gtnqex 7326 . . . . . . . . . . . . . 14  |-  { u  |  A  <Q  u }  e.  _V
1210, 11op1st 6012 . . . . . . . . . . . . 13  |-  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  =  { l  |  l 
<Q  A }
138, 9, 12elab2 2805 . . . . . . . . . . . 12  |-  ( s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  s  <Q  A )
1413biimpi 119 . . . . . . . . . . 11  |-  ( s  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  s  <Q  A )
1514ad2antrl 481 . . . . . . . . . 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 274 . . . . . . . . 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 2663 . . . . . . . . . . . . 13  |-  t  e. 
_V
18 breq1 3902 . . . . . . . . . . . . 13  |-  ( l  =  t  ->  (
l  <Q  B  <->  t  <Q  B ) )
19 ltnqex 7325 . . . . . . . . . . . . . 14  |-  { l  |  l  <Q  B }  e.  _V
20 gtnqex 7326 . . . . . . . . . . . . . 14  |-  { u  |  B  <Q  u }  e.  _V
2119, 20op1st 6012 . . . . . . . . . . . . 13  |-  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  =  { l  |  l 
<Q  B }
2217, 18, 21elab2 2805 . . . . . . . . . . . 12  |-  ( t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  <->  t  <Q  B )
2322biimpi 119 . . . . . . . . . . 11  |-  ( t  e.  ( 1st `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  ->  t  <Q  B )
2423ad2antll 482 . . . . . . . . . 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 274 . . . . . . . . 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 7141 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
2726brel 4561 . . . . . . . . . . 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 4561 . . . . . . . . . . 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 7180 . . . . . . . . . 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 408 . . . . . . . . 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 429 . . . . . . . 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 3902 . . . . . . . . 9  |-  ( r  =  ( s  +Q  t )  ->  (
r  <Q  ( A  +Q  B )  <->  ( s  +Q  t )  <Q  ( A  +Q  B ) ) )
3534adantl 275 . . . . . . . 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 166 . . . . . . 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 2663 . . . . . . . 8  |-  r  e. 
_V
38 breq1 3902 . . . . . . . 8  |-  ( l  =  r  ->  (
l  <Q  ( A  +Q  B )  <->  r  <Q  ( A  +Q  B ) ) )
39 ltnqex 7325 . . . . . . . . 9  |-  { l  |  l  <Q  ( A  +Q  B ) }  e.  _V
40 gtnqex 7326 . . . . . . . . 9  |-  { u  |  ( A  +Q  B )  <Q  u }  e.  _V
4139, 40op1st 6012 . . . . . . . 8  |-  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. )  =  {
l  |  l  <Q 
( A  +Q  B
) }
4237, 38, 41elab2 2805 . . . . . . 7  |-  ( r  e.  ( 1st `  <. { l  |  l  <Q 
( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. )  <->  r  <Q  ( A  +Q  B ) )
4336, 42sylibr 133 . . . . . 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 114 . . . . 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 2534 . . . 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 114 . 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 3073 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 103    <-> wb 104    = wceq 1316    e. wcel 1465   {cab 2103   E.wrex 2394    C_ wss 3041   <.cop 3500   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   1stc1st 6004   Q.cnq 7056    +Q cplq 7058    <Q cltq 7061   P.cnp 7067    +P. cpp 7069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-inp 7242  df-iplp 7244
This theorem is referenced by:  addnqprlemfu  7336  addnqpr  7337
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