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Theorem addclprlem2 8657
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
addclprlem2  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g  +Q  h )  ->  x  e.  ( A  +P.  B ) ) )
Distinct variable groups:    x, g, h    x, A    x, B
Allowed substitution hints:    A( g, h)    B( g, h)

Proof of Theorem addclprlem2
Dummy variables  y 
z  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addclprlem1 8656 . . . . 5  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  x  e.  Q. )  ->  ( x  <Q  ( g  +Q  h )  ->  ( ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  .Q  g )  e.  A
) )
21adantlr 695 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g  +Q  h )  ->  ( ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  .Q  g )  e.  A
) )
3 addclprlem1 8656 . . . . . 6  |-  ( ( ( B  e.  P.  /\  h  e.  B )  /\  x  e.  Q. )  ->  ( x  <Q  ( h  +Q  g )  ->  ( ( x  .Q  ( *Q `  ( h  +Q  g
) ) )  .Q  h )  e.  B
) )
4 addcomnq 8591 . . . . . . 7  |-  ( g  +Q  h )  =  ( h  +Q  g
)
54breq2i 4047 . . . . . 6  |-  ( x 
<Q  ( g  +Q  h
)  <->  x  <Q  ( h  +Q  g ) )
64fveq2i 5544 . . . . . . . . 9  |-  ( *Q
`  ( g  +Q  h ) )  =  ( *Q `  (
h  +Q  g ) )
76oveq2i 5885 . . . . . . . 8  |-  ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  =  ( x  .Q  ( *Q `  ( h  +Q  g ) ) )
87oveq1i 5884 . . . . . . 7  |-  ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  h )  =  ( ( x  .Q  ( *Q `  ( h  +Q  g ) ) )  .Q  h )
98eleq1i 2359 . . . . . 6  |-  ( ( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  h )  e.  B  <->  ( ( x  .Q  ( *Q `  ( h  +Q  g
) ) )  .Q  h )  e.  B
)
103, 5, 93imtr4g 261 . . . . 5  |-  ( ( ( B  e.  P.  /\  h  e.  B )  /\  x  e.  Q. )  ->  ( x  <Q  ( g  +Q  h )  ->  ( ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  .Q  h )  e.  B
) )
1110adantll 694 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g  +Q  h )  ->  ( ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  .Q  h )  e.  B
) )
122, 11jcad 519 . . 3  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g  +Q  h )  ->  ( ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  g )  e.  A  /\  ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  h )  e.  B ) ) )
13 simpl 443 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) ) )
14 simpl 443 . . . . 5  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  A  e.  P. )
15 simpl 443 . . . . 5  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  B  e.  P. )
1614, 15anim12i 549 . . . 4  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( A  e.  P.  /\  B  e.  P. )
)
17 df-plp 8623 . . . . 5  |-  +P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  +Q  z ) } )
18 addclnq 8585 . . . . 5  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
1917, 18genpprecl 8641 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  g )  e.  A  /\  ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  h )  e.  B )  ->  (
( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  g )  +Q  ( ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  .Q  h ) )  e.  ( A  +P.  B
) ) )
2013, 16, 193syl 18 . . 3  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( ( ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  g )  e.  A  /\  ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  h )  e.  B )  ->  (
( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  g )  +Q  ( ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  .Q  h ) )  e.  ( A  +P.  B
) ) )
2112, 20syld 40 . 2  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g  +Q  h )  ->  ( ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  g )  +Q  ( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  h ) )  e.  ( A  +P.  B ) ) )
22 distrnq 8601 . . . . 5  |-  ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  ( g  +Q  h ) )  =  ( ( ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  .Q  g )  +Q  (
( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  h ) )
23 mulassnq 8599 . . . . 5  |-  ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  ( g  +Q  h ) )  =  ( x  .Q  (
( *Q `  (
g  +Q  h ) )  .Q  ( g  +Q  h ) ) )
2422, 23eqtr3i 2318 . . . 4  |-  ( ( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  g )  +Q  ( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  h ) )  =  ( x  .Q  ( ( *Q
`  ( g  +Q  h ) )  .Q  ( g  +Q  h
) ) )
25 mulcomnq 8593 . . . . . . 7  |-  ( ( *Q `  ( g  +Q  h ) )  .Q  ( g  +Q  h ) )  =  ( ( g  +Q  h )  .Q  ( *Q `  ( g  +Q  h ) ) )
26 elprnq 8631 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
27 elprnq 8631 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  h  e.  Q. )
2826, 27anim12i 549 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( g  e.  Q.  /\  h  e.  Q. )
)
29 addclnq 8585 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
30 recidnq 8605 . . . . . . . 8  |-  ( ( g  +Q  h )  e.  Q.  ->  (
( g  +Q  h
)  .Q  ( *Q
`  ( g  +Q  h ) ) )  =  1Q )
3128, 29, 303syl 18 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( ( g  +Q  h )  .Q  ( *Q `  ( g  +Q  h ) ) )  =  1Q )
3225, 31syl5eq 2340 . . . . . 6  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( ( *Q `  ( g  +Q  h
) )  .Q  (
g  +Q  h ) )  =  1Q )
3332oveq2d 5890 . . . . 5  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( x  .Q  (
( *Q `  (
g  +Q  h ) )  .Q  ( g  +Q  h ) ) )  =  ( x  .Q  1Q ) )
34 mulidnq 8603 . . . . 5  |-  ( x  e.  Q.  ->  (
x  .Q  1Q )  =  x )
3533, 34sylan9eq 2348 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( x  .Q  (
( *Q `  (
g  +Q  h ) )  .Q  ( g  +Q  h ) ) )  =  x )
3624, 35syl5eq 2340 . . 3  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( ( ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  .Q  g )  +Q  (
( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  h ) )  =  x )
3736eleq1d 2362 . 2  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( ( ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  g )  +Q  ( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  h ) )  e.  ( A  +P.  B )  <->  x  e.  ( A  +P.  B ) ) )
3821, 37sylibd 205 1  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g  +Q  h )  ->  x  e.  ( A  +P.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Q.cnq 8490   1Qc1q 8491    +Q cplq 8493    .Q cmq 8494   *Qcrq 8495    <Q cltq 8496   P.cnp 8497    +P. cpp 8499
This theorem is referenced by:  addclpr  8658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-plp 8623
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