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Theorem addclprlem1 8594
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 8569 . . 3  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
2 ltrnq 8557 . . . . 5  |-  ( x 
<Q  ( g  +Q  h
)  <->  ( *Q `  ( g  +Q  h
) )  <Q  ( *Q `  x ) )
3 ltmnq 8550 . . . . . 6  |-  ( x  e.  Q.  ->  (
( *Q `  (
g  +Q  h ) )  <Q  ( *Q `  x )  <->  ( x  .Q  ( *Q `  (
g  +Q  h ) ) )  <Q  (
x  .Q  ( *Q
`  x ) ) ) )
4 ovex 5803 . . . . . . 7  |-  ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  e. 
_V
5 ovex 5803 . . . . . . 7  |-  ( x  .Q  ( *Q `  x ) )  e. 
_V
6 ltmnq 8550 . . . . . . 7  |-  ( w  e.  Q.  ->  (
y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
7 vex 2760 . . . . . . 7  |-  g  e. 
_V
8 mulcomnq 8531 . . . . . . 7  |-  ( y  .Q  z )  =  ( z  .Q  y
)
94, 5, 6, 7, 8caovord2 5952 . . . . . 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 8543 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  .Q  ( *Q
`  x ) )  =  1Q )
1312oveq1d 5793 . . . . . 6  |-  ( x  e.  Q.  ->  (
( x  .Q  ( *Q `  x ) )  .Q  g )  =  ( 1Q  .Q  g
) )
14 mulcomnq 8531 . . . . . . 7  |-  ( 1Q 
.Q  g )  =  ( g  .Q  1Q )
15 mulidnq 8541 . . . . . . 7  |-  ( g  e.  Q.  ->  (
g  .Q  1Q )  =  g )
1614, 15syl5eq 2300 . . . . . 6  |-  ( g  e.  Q.  ->  ( 1Q  .Q  g )  =  g )
1713, 16sylan9eqr 2310 . . . . 5  |-  ( ( g  e.  Q.  /\  x  e.  Q. )  ->  ( ( x  .Q  ( *Q `  x ) )  .Q  g )  =  g )
1817breq2d 3995 . . . 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 8571 . . 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 3983   ` cfv 4659  (class class class)co 5778   Q.cnq 8428   1Qc1q 8429    +Q cplq 8431    .Q cmq 8432   *Qcrq 8433    <Q cltq 8434   P.cnp 8435
This theorem is referenced by:  addclprlem2  8595
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 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-omul 6438  df-er 6614  df-ni 8450  df-mi 8452  df-lti 8453  df-mpq 8487  df-ltpq 8488  df-enq 8489  df-nq 8490  df-erq 8491  df-mq 8493  df-1nq 8494  df-rq 8495  df-ltnq 8496  df-np 8559
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