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Theorem addnqprl 7067
Description: Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
Assertion
Ref Expression
addnqprl  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  +Q  H )  ->  X  e.  ( 1st `  ( A  +P.  B
) ) ) )

Proof of Theorem addnqprl
Dummy variables  r  q  s  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7013 . . . . . 6  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 addnqprllem 7065 . . . . . 6  |-  ( ( ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  G  e.  ( 1st `  A ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  +Q  H )  -> 
( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  G )  e.  ( 1st `  A
) ) )
31, 2sylanl1 394 . . . . 5  |-  ( ( ( A  e.  P.  /\  G  e.  ( 1st `  A ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  +Q  H )  -> 
( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  G )  e.  ( 1st `  A
) ) )
43adantlr 461 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  +Q  H )  -> 
( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  G )  e.  ( 1st `  A
) ) )
5 prop 7013 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
6 addnqprllem 7065 . . . . . 6  |-  ( ( ( <. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  H  e.  ( 1st `  B ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  +Q  H )  -> 
( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H )  e.  ( 1st `  B
) ) )
75, 6sylanl1 394 . . . . 5  |-  ( ( ( B  e.  P.  /\  H  e.  ( 1st `  B ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  +Q  H )  -> 
( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H )  e.  ( 1st `  B
) ) )
87adantll 460 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  +Q  H )  -> 
( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H )  e.  ( 1st `  B
) ) )
94, 8jcad 301 . . 3  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  +Q  H )  -> 
( ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  G )  e.  ( 1st `  A )  /\  ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  H )  e.  ( 1st `  B ) ) ) )
10 simpl 107 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) ) )
11 simpl 107 . . . . 5  |-  ( ( A  e.  P.  /\  G  e.  ( 1st `  A ) )  ->  A  e.  P. )
12 simpl 107 . . . . 5  |-  ( ( B  e.  P.  /\  H  e.  ( 1st `  B ) )  ->  B  e.  P. )
1311, 12anim12i 331 . . . 4  |-  ( ( ( A  e.  P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B ) ) )  ->  ( A  e. 
P.  /\  B  e.  P. ) )
14 df-iplp 7006 . . . . 5  |-  +P.  =  ( x  e.  P. ,  y  e.  P.  |->  <. { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
r  e.  ( 1st `  x )  /\  s  e.  ( 1st `  y
)  /\  q  =  ( r  +Q  s
) ) } ,  { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
r  e.  ( 2nd `  x )  /\  s  e.  ( 2nd `  y
)  /\  q  =  ( r  +Q  s
) ) } >. )
15 addclnq 6913 . . . . 5  |-  ( ( r  e.  Q.  /\  s  e.  Q. )  ->  ( r  +Q  s
)  e.  Q. )
1614, 15genpprecll 7052 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( ( X  .Q  ( *Q
`  ( G  +Q  H ) ) )  .Q  G )  e.  ( 1st `  A
)  /\  ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  H )  e.  ( 1st `  B ) )  ->  ( (
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  G )  +Q  ( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H ) )  e.  ( 1st `  ( A  +P.  B
) ) ) )
1710, 13, 163syl 17 . . 3  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( ( ( X  .Q  ( *Q
`  ( G  +Q  H ) ) )  .Q  G )  e.  ( 1st `  A
)  /\  ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  H )  e.  ( 1st `  B ) )  ->  ( (
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  G )  +Q  ( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H ) )  e.  ( 1st `  ( A  +P.  B
) ) ) )
189, 17syld 44 . 2  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  +Q  H )  -> 
( ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  G )  +Q  (
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H ) )  e.  ( 1st `  ( A  +P.  B ) ) ) )
19 simpr 108 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  X  e.  Q. )
20 elprnql 7019 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  G  e.  ( 1st `  A ) )  ->  G  e.  Q. )
211, 20sylan 277 . . . . . . . 8  |-  ( ( A  e.  P.  /\  G  e.  ( 1st `  A ) )  ->  G  e.  Q. )
2221ad2antrr 472 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  G  e.  Q. )
23 elprnql 7019 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  H  e.  ( 1st `  B ) )  ->  H  e.  Q. )
245, 23sylan 277 . . . . . . . 8  |-  ( ( B  e.  P.  /\  H  e.  ( 1st `  B ) )  ->  H  e.  Q. )
2524ad2antlr 473 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  H  e.  Q. )
26 addclnq 6913 . . . . . . 7  |-  ( ( G  e.  Q.  /\  H  e.  Q. )  ->  ( G  +Q  H
)  e.  Q. )
2722, 25, 26syl2anc 403 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( G  +Q  H
)  e.  Q. )
28 recclnq 6930 . . . . . 6  |-  ( ( G  +Q  H )  e.  Q.  ->  ( *Q `  ( G  +Q  H ) )  e. 
Q. )
2927, 28syl 14 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( *Q `  ( G  +Q  H ) )  e.  Q. )
30 mulassnqg 6922 . . . . 5  |-  ( ( X  e.  Q.  /\  ( *Q `  ( G  +Q  H ) )  e.  Q.  /\  ( G  +Q  H )  e. 
Q. )  ->  (
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  ( G  +Q  H ) )  =  ( X  .Q  (
( *Q `  ( G  +Q  H ) )  .Q  ( G  +Q  H ) ) ) )
3119, 29, 27, 30syl3anc 1174 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  ( G  +Q  H ) )  =  ( X  .Q  ( ( *Q `  ( G  +Q  H
) )  .Q  ( G  +Q  H ) ) ) )
32 mulclnq 6914 . . . . . 6  |-  ( ( X  e.  Q.  /\  ( *Q `  ( G  +Q  H ) )  e.  Q. )  -> 
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  e.  Q. )
3319, 29, 32syl2anc 403 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  e.  Q. )
34 distrnqg 6925 . . . . 5  |-  ( ( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  e.  Q.  /\  G  e.  Q.  /\  H  e. 
Q. )  ->  (
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  ( G  +Q  H ) )  =  ( ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  G )  +Q  (
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H ) ) )
3533, 22, 25, 34syl3anc 1174 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  ( G  +Q  H ) )  =  ( ( ( X  .Q  ( *Q
`  ( G  +Q  H ) ) )  .Q  G )  +Q  ( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H ) ) )
36 mulcomnqg 6921 . . . . . . . 8  |-  ( ( ( *Q `  ( G  +Q  H ) )  e.  Q.  /\  ( G  +Q  H )  e. 
Q. )  ->  (
( *Q `  ( G  +Q  H ) )  .Q  ( G  +Q  H ) )  =  ( ( G  +Q  H )  .Q  ( *Q `  ( G  +Q  H ) ) ) )
3729, 27, 36syl2anc 403 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  ( G  +Q  H
) )  .Q  ( G  +Q  H ) )  =  ( ( G  +Q  H )  .Q  ( *Q `  ( G  +Q  H ) ) ) )
38 recidnq 6931 . . . . . . . 8  |-  ( ( G  +Q  H )  e.  Q.  ->  (
( G  +Q  H
)  .Q  ( *Q
`  ( G  +Q  H ) ) )  =  1Q )
3927, 38syl 14 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  +Q  H )  .Q  ( *Q `  ( G  +Q  H ) ) )  =  1Q )
4037, 39eqtrd 2120 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  ( G  +Q  H
) )  .Q  ( G  +Q  H ) )  =  1Q )
4140oveq2d 5650 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  (
( *Q `  ( G  +Q  H ) )  .Q  ( G  +Q  H ) ) )  =  ( X  .Q  1Q ) )
42 mulidnq 6927 . . . . . 6  |-  ( X  e.  Q.  ->  ( X  .Q  1Q )  =  X )
4342adantl 271 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  1Q )  =  X )
4441, 43eqtrd 2120 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  (
( *Q `  ( G  +Q  H ) )  .Q  ( G  +Q  H ) ) )  =  X )
4531, 35, 443eqtr3d 2128 . . 3  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  G )  +Q  (
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H ) )  =  X )
4645eleq1d 2156 . 2  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( ( ( X  .Q  ( *Q
`  ( G  +Q  H ) ) )  .Q  G )  +Q  ( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H ) )  e.  ( 1st `  ( A  +P.  B
) )  <->  X  e.  ( 1st `  ( A  +P.  B ) ) ) )
4718, 46sylibd 147 1  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  +Q  H )  ->  X  e.  ( 1st `  ( A  +P.  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   <.cop 3444   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   1stc1st 5891   2ndc2nd 5892   Q.cnq 6818   1Qc1q 6819    +Q cplq 6820    .Q cmq 6821   *Qcrq 6822    <Q cltq 6823   P.cnp 6829    +P. cpp 6831
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-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-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-inp 7004  df-iplp 7006
This theorem is referenced by:  addlocprlemlt  7069  addclpr  7075
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