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Theorem addclprlem1 8520
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 8495 . . 3  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
2 ltrnq 8483 . . . . 5  |-  ( x 
<Q  ( g  +Q  h
)  <->  ( *Q `  ( g  +Q  h
) )  <Q  ( *Q `  x ) )
3 ltmnq 8476 . . . . . 6  |-  ( x  e.  Q.  ->  (
( *Q `  (
g  +Q  h ) )  <Q  ( *Q `  x )  <->  ( x  .Q  ( *Q `  (
g  +Q  h ) ) )  <Q  (
x  .Q  ( *Q
`  x ) ) ) )
4 ovex 5735 . . . . . . 7  |-  ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  e. 
_V
5 ovex 5735 . . . . . . 7  |-  ( x  .Q  ( *Q `  x ) )  e. 
_V
6 ltmnq 8476 . . . . . . 7  |-  ( w  e.  Q.  ->  (
y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
7 vex 2730 . . . . . . 7  |-  g  e. 
_V
8 mulcomnq 8457 . . . . . . 7  |-  ( y  .Q  z )  =  ( z  .Q  y
)
94, 5, 6, 7, 8caovord2 5884 . . . . . 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 8469 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  .Q  ( *Q
`  x ) )  =  1Q )
1312oveq1d 5725 . . . . . 6  |-  ( x  e.  Q.  ->  (
( x  .Q  ( *Q `  x ) )  .Q  g )  =  ( 1Q  .Q  g
) )
14 mulcomnq 8457 . . . . . . 7  |-  ( 1Q 
.Q  g )  =  ( g  .Q  1Q )
15 mulidnq 8467 . . . . . . 7  |-  ( g  e.  Q.  ->  (
g  .Q  1Q )  =  g )
1614, 15syl5eq 2297 . . . . . 6  |-  ( g  e.  Q.  ->  ( 1Q  .Q  g )  =  g )
1713, 16sylan9eqr 2307 . . . . 5  |-  ( ( g  e.  Q.  /\  x  e.  Q. )  ->  ( ( x  .Q  ( *Q `  x ) )  .Q  g )  =  g )
1817breq2d 3932 . . . 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 8497 . . 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 3920   ` cfv 4592  (class class class)co 5710   Q.cnq 8354   1Qc1q 8355    +Q cplq 8357    .Q cmq 8358   *Qcrq 8359    <Q cltq 8360   P.cnp 8361
This theorem is referenced by:  addclprlem2  8521
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-omul 6370  df-er 6546  df-ni 8376  df-mi 8378  df-lti 8379  df-mpq 8413  df-ltpq 8414  df-enq 8415  df-nq 8416  df-erq 8417  df-mq 8419  df-1nq 8420  df-rq 8421  df-ltnq 8422  df-np 8485
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