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Theorem addclprlem1 8642
Description: Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
Assertion
Ref Expression
addclprlem1  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  x  e.  Q. )  ->  ( x  <Q  ( g  +Q  h )  ->  ( ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  .Q  g )  e.  A
) )

Proof of Theorem addclprlem1
Dummy variables  y 
z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elprnq 8617 . . 3  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
2 ltrnq 8605 . . . . 5  |-  ( x 
<Q  ( g  +Q  h
)  <->  ( *Q `  ( g  +Q  h
) )  <Q  ( *Q `  x ) )
3 ltmnq 8598 . . . . . 6  |-  ( x  e.  Q.  ->  (
( *Q `  (
g  +Q  h ) )  <Q  ( *Q `  x )  <->  ( x  .Q  ( *Q `  (
g  +Q  h ) ) )  <Q  (
x  .Q  ( *Q
`  x ) ) ) )
4 ovex 5885 . . . . . . 7  |-  ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  e. 
_V
5 ovex 5885 . . . . . . 7  |-  ( x  .Q  ( *Q `  x ) )  e. 
_V
6 ltmnq 8598 . . . . . . 7  |-  ( w  e.  Q.  ->  (
y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
7 vex 2793 . . . . . . 7  |-  g  e. 
_V
8 mulcomnq 8579 . . . . . . 7  |-  ( y  .Q  z )  =  ( z  .Q  y
)
94, 5, 6, 7, 8caovord2 6034 . . . . . 6  |-  ( g  e.  Q.  ->  (
( x  .Q  ( *Q `  ( g  +Q  h ) ) ) 
<Q  ( x  .Q  ( *Q `  x ) )  <-> 
( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  g ) 
<Q  ( ( x  .Q  ( *Q `  x ) )  .Q  g ) ) )
103, 9sylan9bbr 681 . . . . 5  |-  ( ( g  e.  Q.  /\  x  e.  Q. )  ->  ( ( *Q `  ( g  +Q  h
) )  <Q  ( *Q `  x )  <->  ( (
x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  g )  <Q 
( ( x  .Q  ( *Q `  x ) )  .Q  g ) ) )
112, 10syl5bb 248 . . . 4  |-  ( ( g  e.  Q.  /\  x  e.  Q. )  ->  ( x  <Q  (
g  +Q  h )  <-> 
( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  g ) 
<Q  ( ( x  .Q  ( *Q `  x ) )  .Q  g ) ) )
12 recidnq 8591 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  .Q  ( *Q
`  x ) )  =  1Q )
1312oveq1d 5875 . . . . . 6  |-  ( x  e.  Q.  ->  (
( x  .Q  ( *Q `  x ) )  .Q  g )  =  ( 1Q  .Q  g
) )
14 mulcomnq 8579 . . . . . . 7  |-  ( 1Q 
.Q  g )  =  ( g  .Q  1Q )
15 mulidnq 8589 . . . . . . 7  |-  ( g  e.  Q.  ->  (
g  .Q  1Q )  =  g )
1614, 15syl5eq 2329 . . . . . 6  |-  ( g  e.  Q.  ->  ( 1Q  .Q  g )  =  g )
1713, 16sylan9eqr 2339 . . . . 5  |-  ( ( g  e.  Q.  /\  x  e.  Q. )  ->  ( ( x  .Q  ( *Q `  x ) )  .Q  g )  =  g )
1817breq2d 4037 . . . 4  |-  ( ( g  e.  Q.  /\  x  e.  Q. )  ->  ( ( ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  .Q  g )  <Q  (
( x  .Q  ( *Q `  x ) )  .Q  g )  <->  ( (
x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  g )  <Q 
g ) )
1911, 18bitrd 244 . . 3  |-  ( ( g  e.  Q.  /\  x  e.  Q. )  ->  ( x  <Q  (
g  +Q  h )  <-> 
( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  g ) 
<Q  g ) )
201, 19sylan 457 . 2  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  x  e.  Q. )  ->  ( x  <Q  ( g  +Q  h )  <-> 
( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  g ) 
<Q  g ) )
21 prcdnq 8619 . . 3  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  ( ( ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  .Q  g )  <Q  g  ->  ( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  g )  e.  A ) )
2221adantr 451 . 2  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  x  e.  Q. )  ->  ( ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  g )  <Q 
g  ->  ( (
x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  g )  e.  A ) )
2320, 22sylbid 206 1  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  x  e.  Q. )  ->  ( x  <Q  ( g  +Q  h )  ->  ( ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  .Q  g )  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1686   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   Q.cnq 8476   1Qc1q 8477    +Q cplq 8479    .Q cmq 8480   *Qcrq 8481    <Q cltq 8482   P.cnp 8483
This theorem is referenced by:  addclprlem2  8643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-omul 6486  df-er 6662  df-ni 8498  df-mi 8500  df-lti 8501  df-mpq 8535  df-ltpq 8536  df-enq 8537  df-nq 8538  df-erq 8539  df-mq 8541  df-1nq 8542  df-rq 8543  df-ltnq 8544  df-np 8607
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