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Theorem addclprlem1 8893
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 8868 . . 3  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
2 ltrnq 8856 . . . . 5  |-  ( x 
<Q  ( g  +Q  h
)  <->  ( *Q `  ( g  +Q  h
) )  <Q  ( *Q `  x ) )
3 ltmnq 8849 . . . . . 6  |-  ( x  e.  Q.  ->  (
( *Q `  (
g  +Q  h ) )  <Q  ( *Q `  x )  <->  ( x  .Q  ( *Q `  (
g  +Q  h ) ) )  <Q  (
x  .Q  ( *Q
`  x ) ) ) )
4 ovex 6106 . . . . . . 7  |-  ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  e. 
_V
5 ovex 6106 . . . . . . 7  |-  ( x  .Q  ( *Q `  x ) )  e. 
_V
6 ltmnq 8849 . . . . . . 7  |-  ( w  e.  Q.  ->  (
y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
7 vex 2959 . . . . . . 7  |-  g  e. 
_V
8 mulcomnq 8830 . . . . . . 7  |-  ( y  .Q  z )  =  ( z  .Q  y
)
94, 5, 6, 7, 8caovord2 6259 . . . . . 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 682 . . . . 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 249 . . . 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 8842 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  .Q  ( *Q
`  x ) )  =  1Q )
1312oveq1d 6096 . . . . . 6  |-  ( x  e.  Q.  ->  (
( x  .Q  ( *Q `  x ) )  .Q  g )  =  ( 1Q  .Q  g
) )
14 mulcomnq 8830 . . . . . . 7  |-  ( 1Q 
.Q  g )  =  ( g  .Q  1Q )
15 mulidnq 8840 . . . . . . 7  |-  ( g  e.  Q.  ->  (
g  .Q  1Q )  =  g )
1614, 15syl5eq 2480 . . . . . 6  |-  ( g  e.  Q.  ->  ( 1Q  .Q  g )  =  g )
1713, 16sylan9eqr 2490 . . . . 5  |-  ( ( g  e.  Q.  /\  x  e.  Q. )  ->  ( ( x  .Q  ( *Q `  x ) )  .Q  g )  =  g )
1817breq2d 4224 . . . 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 245 . . 3  |-  ( ( g  e.  Q.  /\  x  e.  Q. )  ->  ( x  <Q  (
g  +Q  h )  <-> 
( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  g ) 
<Q  g ) )
201, 19sylan 458 . 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 8870 . . 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 452 . 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 207 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 177    /\ wa 359    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Q.cnq 8727   1Qc1q 8728    +Q cplq 8730    .Q cmq 8731   *Qcrq 8732    <Q cltq 8733   P.cnp 8734
This theorem is referenced by:  addclprlem2  8894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-ni 8749  df-mi 8751  df-lti 8752  df-mpq 8786  df-ltpq 8787  df-enq 8788  df-nq 8789  df-erq 8790  df-mq 8792  df-1nq 8793  df-rq 8794  df-ltnq 8795  df-np 8858
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