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Theorem addnqprlemru 7330
Description: Lemma for addnqpr 7333. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
Assertion
Ref Expression
addnqprlemru  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  C_  ( 2nd ` 
<. { l  |  l 
<Q  ( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. ) )
Distinct variable groups:    A, l, u    B, l, u

Proof of Theorem addnqprlemru
Dummy variables  f  g  h  r  s  t  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqprlu 7319 . . . . . 6  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
2 nqprlu 7319 . . . . . 6  |-  ( B  e.  Q.  ->  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >.  e.  P. )
3 df-iplp 7240 . . . . . . 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 7147 . . . . . . 7  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
53, 4genpelvu 7285 . . . . . 6  |-  ( (
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  e.  P.  /\ 
<. { l  |  l 
<Q  B } ,  {
u  |  B  <Q  u } >.  e.  P. )  ->  ( r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  <->  E. s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
r  =  ( s  +Q  t ) ) )
61, 2, 5syl2an 285 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  <->  E. s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
r  =  ( s  +Q  t ) ) )
76biimpa 292 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )  ->  E. s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
r  =  ( s  +Q  t ) )
8 vex 2661 . . . . . . . . . . . . 13  |-  s  e. 
_V
9 breq2 3901 . . . . . . . . . . . . 13  |-  ( u  =  s  ->  ( A  <Q  u  <->  A  <Q  s ) )
10 ltnqex 7321 . . . . . . . . . . . . . 14  |-  { l  |  l  <Q  A }  e.  _V
11 gtnqex 7322 . . . . . . . . . . . . . 14  |-  { u  |  A  <Q  u }  e.  _V
1210, 11op2nd 6011 . . . . . . . . . . . . 13  |-  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  =  { u  |  A  <Q  u }
138, 9, 12elab2 2803 . . . . . . . . . . . 12  |-  ( s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  A  <Q  s )
1413biimpi 119 . . . . . . . . . . 11  |-  ( s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  A  <Q  s )
1514ad2antrl 479 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  ->  A  <Q  s )
1615adantr 272 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  A  <Q  s )
17 vex 2661 . . . . . . . . . . . . 13  |-  t  e. 
_V
18 breq2 3901 . . . . . . . . . . . . 13  |-  ( u  =  t  ->  ( B  <Q  u  <->  B  <Q  t ) )
19 ltnqex 7321 . . . . . . . . . . . . . 14  |-  { l  |  l  <Q  B }  e.  _V
20 gtnqex 7322 . . . . . . . . . . . . . 14  |-  { u  |  B  <Q  u }  e.  _V
2119, 20op2nd 6011 . . . . . . . . . . . . 13  |-  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  =  { u  |  B  <Q  u }
2217, 18, 21elab2 2803 . . . . . . . . . . . 12  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  <->  B  <Q  t )
2322biimpi 119 . . . . . . . . . . 11  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )  ->  B  <Q  t )
2423ad2antll 480 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  ->  B  <Q  t )
2524adantr 272 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  B  <Q  t )
26 ltrelnq 7137 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
2726brel 4559 . . . . . . . . . . 11  |-  ( A 
<Q  s  ->  ( A  e.  Q.  /\  s  e.  Q. ) )
2816, 27syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( A  e.  Q.  /\  s  e.  Q. )
)
2926brel 4559 . . . . . . . . . . 11  |-  ( B 
<Q  t  ->  ( B  e.  Q.  /\  t  e.  Q. ) )
3025, 29syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( B  e.  Q.  /\  t  e.  Q. )
)
31 lt2addnq 7176 . . . . . . . . . 10  |-  ( ( ( A  e.  Q.  /\  s  e.  Q. )  /\  ( B  e.  Q.  /\  t  e.  Q. )
)  ->  ( ( A  <Q  s  /\  B  <Q  t )  ->  ( A  +Q  B )  <Q 
( s  +Q  t
) ) )
3228, 30, 31syl2anc 406 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( ( A  <Q  s  /\  B  <Q  t
)  ->  ( A  +Q  B )  <Q  (
s  +Q  t ) ) )
3316, 25, 32mp2and 427 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( A  +Q  B
)  <Q  ( s  +Q  t ) )
34 breq2 3901 . . . . . . . . 9  |-  ( r  =  ( s  +Q  t )  ->  (
( A  +Q  B
)  <Q  r  <->  ( A  +Q  B )  <Q  (
s  +Q  t ) ) )
3534adantl 273 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( ( A  +Q  B )  <Q  r  <->  ( A  +Q  B ) 
<Q  ( s  +Q  t
) ) )
3633, 35mpbird 166 . . . . . . 7  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( A  +Q  B
)  <Q  r )
37 vex 2661 . . . . . . . 8  |-  r  e. 
_V
38 breq2 3901 . . . . . . . 8  |-  ( u  =  r  ->  (
( A  +Q  B
)  <Q  u  <->  ( A  +Q  B )  <Q  r
) )
39 ltnqex 7321 . . . . . . . . 9  |-  { l  |  l  <Q  ( A  +Q  B ) }  e.  _V
40 gtnqex 7322 . . . . . . . . 9  |-  { u  |  ( A  +Q  B )  <Q  u }  e.  _V
4139, 40op2nd 6011 . . . . . . . 8  |-  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. )  =  {
u  |  ( A  +Q  B )  <Q  u }
4237, 38, 41elab2 2803 . . . . . . 7  |-  ( r  e.  ( 2nd `  <. { l  |  l  <Q 
( A  +Q  B
) } ,  {
u  |  ( A  +Q  B )  <Q  u } >. )  <->  ( A  +Q  B )  <Q  r
)
4336, 42sylibr 133 . . . . . 6  |-  ( ( ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. ) )
4443ex 114 . . . . 5  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
) )  ->  (
r  =  ( s  +Q  t )  -> 
r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. ) ) )
4544rexlimdvva 2532 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )  -> 
( E. s  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) r  =  ( s  +Q  t )  ->  r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B
)  <Q  u } >. ) ) )
467, 45mpd 13 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  r  e.  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )  -> 
r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >. ) )
4746ex 114 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( r  e.  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  ->  r  e.  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B
)  <Q  u } >. ) ) )
4847ssrdv 3071 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( 2nd `  ( <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
)  C_  ( 2nd ` 
<. { 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 1314    e. wcel 1463   {cab 2101   E.wrex 2392    C_ wss 3039   <.cop 3498   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   2ndc2nd 6003   Q.cnq 7052    +Q cplq 7054    <Q cltq 7057   P.cnp 7063    +P. cpp 7065
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-inp 7238  df-iplp 7240
This theorem is referenced by:  addnqprlemfl  7331  addnqpr  7333
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