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Theorem addnqprl 6685
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 6631 . . . . . 6  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 addnqprllem 6683 . . . . . 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 388 . . . . 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 454 . . . 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 6631 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
6 addnqprllem 6683 . . . . . 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 388 . . . . 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 453 . . . 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 295 . . 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 106 . . . 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 106 . . . . 5  |-  ( ( A  e.  P.  /\  G  e.  ( 1st `  A ) )  ->  A  e.  P. )
12 simpl 106 . . . . 5  |-  ( ( B  e.  P.  /\  H  e.  ( 1st `  B ) )  ->  B  e.  P. )
1311, 12anim12i 325 . . . 4  |-  ( ( ( A  e.  P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B ) ) )  ->  ( A  e. 
P.  /\  B  e.  P. ) )
14 df-iplp 6624 . . . . 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 6531 . . . . 5  |-  ( ( r  e.  Q.  /\  s  e.  Q. )  ->  ( r  +Q  s
)  e.  Q. )
1614, 15genpprecll 6670 . . . 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 107 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  X  e.  Q. )
20 elprnql 6637 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  G  e.  ( 1st `  A ) )  ->  G  e.  Q. )
211, 20sylan 271 . . . . . . . 8  |-  ( ( A  e.  P.  /\  G  e.  ( 1st `  A ) )  ->  G  e.  Q. )
2221ad2antrr 465 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  G  e.  Q. )
23 elprnql 6637 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  H  e.  ( 1st `  B ) )  ->  H  e.  Q. )
245, 23sylan 271 . . . . . . . 8  |-  ( ( B  e.  P.  /\  H  e.  ( 1st `  B ) )  ->  H  e.  Q. )
2524ad2antlr 466 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  H  e.  Q. )
26 addclnq 6531 . . . . . . 7  |-  ( ( G  e.  Q.  /\  H  e.  Q. )  ->  ( G  +Q  H
)  e.  Q. )
2722, 25, 26syl2anc 397 . . . . . 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 6548 . . . . . 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 6540 . . . . 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 1146 . . . 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 6532 . . . . . 6  |-  ( ( X  e.  Q.  /\  ( *Q `  ( G  +Q  H ) )  e.  Q. )  -> 
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  e.  Q. )
3319, 29, 32syl2anc 397 . . . . 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 6543 . . . . 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 1146 . . . 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 6539 . . . . . . . 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 397 . . . . . . 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 6549 . . . . . . . 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 2088 . . . . . 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 5556 . . . . 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 6545 . . . . . 6  |-  ( X  e.  Q.  ->  ( X  .Q  1Q )  =  X )
4342adantl 266 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  1Q )  =  X )
4441, 43eqtrd 2088 . . . 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 2096 . . 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 2122 . 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 142 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 101    = wceq 1259    e. wcel 1409   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436   1Qc1q 6437    +Q cplq 6438    .Q cmq 6439   *Qcrq 6440    <Q cltq 6441   P.cnp 6447    +P. cpp 6449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-inp 6622  df-iplp 6624
This theorem is referenced by:  addlocprlemlt  6687  addclpr  6693
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