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Theorem addnqpru 7186
Description: Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
Assertion
Ref Expression
addnqpru  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  +Q  H )  <Q  X  ->  X  e.  ( 2nd `  ( A  +P.  B
) ) ) )

Proof of Theorem addnqpru
Dummy variables  r  q  s  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7131 . . . . . 6  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 addnqprulem 7184 . . . . . 6  |-  ( ( ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  G  e.  ( 2nd `  A ) )  /\  X  e.  Q. )  ->  ( ( G  +Q  H )  <Q  X  -> 
( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  G )  e.  ( 2nd `  A
) ) )
31, 2sylanl1 395 . . . . 5  |-  ( ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  /\  X  e.  Q. )  ->  ( ( G  +Q  H )  <Q  X  -> 
( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  G )  e.  ( 2nd `  A
) ) )
43adantlr 462 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  +Q  H )  <Q  X  -> 
( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  G )  e.  ( 2nd `  A
) ) )
5 prop 7131 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
6 addnqprulem 7184 . . . . . 6  |-  ( ( ( <. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  H  e.  ( 2nd `  B ) )  /\  X  e.  Q. )  ->  ( ( G  +Q  H )  <Q  X  -> 
( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H )  e.  ( 2nd `  B
) ) )
75, 6sylanl1 395 . . . . 5  |-  ( ( ( B  e.  P.  /\  H  e.  ( 2nd `  B ) )  /\  X  e.  Q. )  ->  ( ( G  +Q  H )  <Q  X  -> 
( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H )  e.  ( 2nd `  B
) ) )
87adantll 461 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  +Q  H )  <Q  X  -> 
( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H )  e.  ( 2nd `  B
) ) )
94, 8jcad 302 . . 3  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  +Q  H )  <Q  X  -> 
( ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  G )  e.  ( 2nd `  A )  /\  ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  H )  e.  ( 2nd `  B ) ) ) )
10 simpl 108 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) ) )
11 simpl 108 . . . . 5  |-  ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  ->  A  e.  P. )
12 simpl 108 . . . . 5  |-  ( ( B  e.  P.  /\  H  e.  ( 2nd `  B ) )  ->  B  e.  P. )
1311, 12anim12i 332 . . . 4  |-  ( ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B ) ) )  ->  ( A  e. 
P.  /\  B  e.  P. ) )
14 df-iplp 7124 . . . . 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 7031 . . . . 5  |-  ( ( r  e.  Q.  /\  s  e.  Q. )  ->  ( r  +Q  s
)  e.  Q. )
1614, 15genppreclu 7171 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( ( X  .Q  ( *Q
`  ( G  +Q  H ) ) )  .Q  G )  e.  ( 2nd `  A
)  /\  ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  H )  e.  ( 2nd `  B ) )  ->  ( (
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  G )  +Q  ( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H ) )  e.  ( 2nd `  ( A  +P.  B
) ) ) )
1710, 13, 163syl 17 . . 3  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( ( ( X  .Q  ( *Q
`  ( G  +Q  H ) ) )  .Q  G )  e.  ( 2nd `  A
)  /\  ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  H )  e.  ( 2nd `  B ) )  ->  ( (
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  G )  +Q  ( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H ) )  e.  ( 2nd `  ( A  +P.  B
) ) ) )
189, 17syld 45 . 2  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  +Q  H )  <Q  X  -> 
( ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  G )  +Q  (
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H ) )  e.  ( 2nd `  ( A  +P.  B ) ) ) )
19 simpr 109 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  X  e.  Q. )
20 elprnqu 7138 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  G  e.  ( 2nd `  A ) )  ->  G  e.  Q. )
211, 20sylan 278 . . . . . . . 8  |-  ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  ->  G  e.  Q. )
2221ad2antrr 473 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  G  e.  Q. )
23 elprnqu 7138 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  H  e.  ( 2nd `  B ) )  ->  H  e.  Q. )
245, 23sylan 278 . . . . . . . 8  |-  ( ( B  e.  P.  /\  H  e.  ( 2nd `  B ) )  ->  H  e.  Q. )
2524ad2antlr 474 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  H  e.  Q. )
26 addclnq 7031 . . . . . . 7  |-  ( ( G  e.  Q.  /\  H  e.  Q. )  ->  ( G  +Q  H
)  e.  Q. )
2722, 25, 26syl2anc 404 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( G  +Q  H
)  e.  Q. )
28 recclnq 7048 . . . . . 6  |-  ( ( G  +Q  H )  e.  Q.  ->  ( *Q `  ( G  +Q  H ) )  e. 
Q. )
2927, 28syl 14 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( *Q `  ( G  +Q  H ) )  e.  Q. )
30 mulassnqg 7040 . . . . 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 1181 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  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 7032 . . . . . 6  |-  ( ( X  e.  Q.  /\  ( *Q `  ( G  +Q  H ) )  e.  Q. )  -> 
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  e.  Q. )
3319, 29, 32syl2anc 404 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  e.  Q. )
34 distrnqg 7043 . . . . 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 1181 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  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 7039 . . . . . . . 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 404 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  ( G  +Q  H
) )  .Q  ( G  +Q  H ) )  =  ( ( G  +Q  H )  .Q  ( *Q `  ( G  +Q  H ) ) ) )
38 recidnq 7049 . . . . . . . 8  |-  ( ( G  +Q  H )  e.  Q.  ->  (
( G  +Q  H
)  .Q  ( *Q
`  ( G  +Q  H ) ) )  =  1Q )
3927, 38syl 14 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  +Q  H )  .Q  ( *Q `  ( G  +Q  H ) ) )  =  1Q )
4037, 39eqtrd 2127 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  ( G  +Q  H
) )  .Q  ( G  +Q  H ) )  =  1Q )
4140oveq2d 5706 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  (
( *Q `  ( G  +Q  H ) )  .Q  ( G  +Q  H ) ) )  =  ( X  .Q  1Q ) )
42 mulidnq 7045 . . . . . 6  |-  ( X  e.  Q.  ->  ( X  .Q  1Q )  =  X )
4342adantl 272 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  1Q )  =  X )
4441, 43eqtrd 2127 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  (
( *Q `  ( G  +Q  H ) )  .Q  ( G  +Q  H ) ) )  =  X )
4531, 35, 443eqtr3d 2135 . . 3  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( ( X  .Q  ( *Q `  ( G  +Q  H
) ) )  .Q  G )  +Q  (
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H ) )  =  X )
4645eleq1d 2163 . 2  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( ( ( X  .Q  ( *Q
`  ( G  +Q  H ) ) )  .Q  G )  +Q  ( ( X  .Q  ( *Q `  ( G  +Q  H ) ) )  .Q  H ) )  e.  ( 2nd `  ( A  +P.  B
) )  <->  X  e.  ( 2nd `  ( A  +P.  B ) ) ) )
4718, 46sylibd 148 1  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  +Q  H )  <Q  X  ->  X  e.  ( 2nd `  ( A  +P.  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1296    e. wcel 1445   <.cop 3469   class class class wbr 3867   ` cfv 5049  (class class class)co 5690   1stc1st 5947   2ndc2nd 5948   Q.cnq 6936   1Qc1q 6937    +Q cplq 6938    .Q cmq 6939   *Qcrq 6940    <Q cltq 6941   P.cnp 6947    +P. cpp 6949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009  df-inp 7122  df-iplp 7124
This theorem is referenced by:  addlocprlemeq  7189  addlocprlemgt  7190  addclpr  7193
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