ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  addnqpru Unicode version

Theorem addnqpru 7352
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 7297 . . . . . 6  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 addnqprulem 7350 . . . . . 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 399 . . . . 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 468 . . . 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 7297 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
6 addnqprulem 7350 . . . . . 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 399 . . . . 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 467 . . . 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 305 . . 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 336 . . . 4  |-  ( ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B ) ) )  ->  ( A  e. 
P.  /\  B  e.  P. ) )
14 df-iplp 7290 . . . . 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 7197 . . . . 5  |-  ( ( r  e.  Q.  /\  s  e.  Q. )  ->  ( r  +Q  s
)  e.  Q. )
1614, 15genppreclu 7337 . . . 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 7304 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  G  e.  ( 2nd `  A ) )  ->  G  e.  Q. )
211, 20sylan 281 . . . . . . . 8  |-  ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  ->  G  e.  Q. )
2221ad2antrr 479 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  G  e.  Q. )
23 elprnqu 7304 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  H  e.  ( 2nd `  B ) )  ->  H  e.  Q. )
245, 23sylan 281 . . . . . . . 8  |-  ( ( B  e.  P.  /\  H  e.  ( 2nd `  B ) )  ->  H  e.  Q. )
2524ad2antlr 480 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  H  e.  Q. )
26 addclnq 7197 . . . . . . 7  |-  ( ( G  e.  Q.  /\  H  e.  Q. )  ->  ( G  +Q  H
)  e.  Q. )
2722, 25, 26syl2anc 408 . . . . . 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 7214 . . . . . 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 7206 . . . . 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 1216 . . . 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 7198 . . . . . 6  |-  ( ( X  e.  Q.  /\  ( *Q `  ( G  +Q  H ) )  e.  Q. )  -> 
( X  .Q  ( *Q `  ( G  +Q  H ) ) )  e.  Q. )
3319, 29, 32syl2anc 408 . . . . 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 7209 . . . . 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 1216 . . . 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 7205 . . . . . . . 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 408 . . . . . . 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 7215 . . . . . . . 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 2172 . . . . . 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 5790 . . . . 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 7211 . . . . . 6  |-  ( X  e.  Q.  ->  ( X  .Q  1Q )  =  X )
4342adantl 275 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  1Q )  =  X )
4441, 43eqtrd 2172 . . . 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 2180 . . 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 2208 . 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 1331    e. wcel 1480   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7102   1Qc1q 7103    +Q cplq 7104    .Q cmq 7105   *Qcrq 7106    <Q cltq 7107   P.cnp 7113    +P. cpp 7115
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7126  df-pli 7127  df-mi 7128  df-lti 7129  df-plpq 7166  df-mpq 7167  df-enq 7169  df-nqqs 7170  df-plqqs 7171  df-mqqs 7172  df-1nqqs 7173  df-rq 7174  df-ltnqqs 7175  df-inp 7288  df-iplp 7290
This theorem is referenced by:  addlocprlemeq  7355  addlocprlemgt  7356  addclpr  7359
  Copyright terms: Public domain W3C validator