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Theorem addclprlem1 8608
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
StepHypRef Expression
1 elprnq 8583 . . 3  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
2 ltrnq 8571 . . . . 5  |-  ( x 
<Q  ( g  +Q  h
)  <->  ( *Q `  ( g  +Q  h
) )  <Q  ( *Q `  x ) )
3 ltmnq 8564 . . . . . 6  |-  ( x  e.  Q.  ->  (
( *Q `  (
g  +Q  h ) )  <Q  ( *Q `  x )  <->  ( x  .Q  ( *Q `  (
g  +Q  h ) ) )  <Q  (
x  .Q  ( *Q
`  x ) ) ) )
4 ovex 5817 . . . . . . 7  |-  ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  e. 
_V
5 ovex 5817 . . . . . . 7  |-  ( x  .Q  ( *Q `  x ) )  e. 
_V
6 ltmnq 8564 . . . . . . 7  |-  ( w  e.  Q.  ->  (
y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
7 vex 2766 . . . . . . 7  |-  g  e. 
_V
8 mulcomnq 8545 . . . . . . 7  |-  ( y  .Q  z )  =  ( z  .Q  y
)
94, 5, 6, 7, 8caovord2 5966 . . . . . 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 684 . . . . 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 250 . . . 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 8557 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  .Q  ( *Q
`  x ) )  =  1Q )
1312oveq1d 5807 . . . . . 6  |-  ( x  e.  Q.  ->  (
( x  .Q  ( *Q `  x ) )  .Q  g )  =  ( 1Q  .Q  g
) )
14 mulcomnq 8545 . . . . . . 7  |-  ( 1Q 
.Q  g )  =  ( g  .Q  1Q )
15 mulidnq 8555 . . . . . . 7  |-  ( g  e.  Q.  ->  (
g  .Q  1Q )  =  g )
1614, 15syl5eq 2302 . . . . . 6  |-  ( g  e.  Q.  ->  ( 1Q  .Q  g )  =  g )
1713, 16sylan9eqr 2312 . . . . 5  |-  ( ( g  e.  Q.  /\  x  e.  Q. )  ->  ( ( x  .Q  ( *Q `  x ) )  .Q  g )  =  g )
1817breq2d 4009 . . . 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 246 . . 3  |-  ( ( g  e.  Q.  /\  x  e.  Q. )  ->  ( x  <Q  (
g  +Q  h )  <-> 
( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  g ) 
<Q  g ) )
201, 19sylan 459 . 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 8585 . . 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 453 . 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 208 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 6    <-> wb 178    /\ wa 360    e. wcel 1621   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Q.cnq 8442   1Qc1q 8443    +Q cplq 8445    .Q cmq 8446   *Qcrq 8447    <Q cltq 8448   P.cnp 8449
This theorem is referenced by:  addclprlem2  8609
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-omul 6452  df-er 6628  df-ni 8464  df-mi 8466  df-lti 8467  df-mpq 8501  df-ltpq 8502  df-enq 8503  df-nq 8504  df-erq 8505  df-mq 8507  df-1nq 8508  df-rq 8509  df-ltnq 8510  df-np 8573
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