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Theorem addclprlem2 9298
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 9297 . . . . 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 714 . . . 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 9297 . . . . . 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 9232 . . . . . . 7  |-  ( g  +Q  h )  =  ( h  +Q  g
)
54breq2i 4409 . . . . . 6  |-  ( x 
<Q  ( g  +Q  h
)  <->  x  <Q  ( h  +Q  g ) )
64fveq2i 5803 . . . . . . . . 9  |-  ( *Q
`  ( g  +Q  h ) )  =  ( *Q `  (
h  +Q  g ) )
76oveq2i 6212 . . . . . . . 8  |-  ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  =  ( x  .Q  ( *Q `  ( h  +Q  g ) ) )
87oveq1i 6211 . . . . . . 7  |-  ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  h )  =  ( ( x  .Q  ( *Q `  ( h  +Q  g ) ) )  .Q  h )
98eleq1i 2531 . . . . . 6  |-  ( ( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  h )  e.  B  <->  ( ( x  .Q  ( *Q `  ( h  +Q  g
) ) )  .Q  h )  e.  B
)
103, 5, 93imtr4g 270 . . . . 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 713 . . . 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 533 . . 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 457 . . . 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 457 . . . . 5  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  A  e.  P. )
15 simpl 457 . . . . 5  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  B  e.  P. )
1614, 15anim12i 566 . . . 4  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( A  e.  P.  /\  B  e.  P. )
)
17 df-plp 9264 . . . . 5  |-  +P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  +Q  z ) } )
18 addclnq 9226 . . . . 5  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
1917, 18genpprecl 9282 . . . 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 20 . . 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 44 . 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 9242 . . . . 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 9240 . . . . 5  |-  ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  ( g  +Q  h ) )  =  ( x  .Q  (
( *Q `  (
g  +Q  h ) )  .Q  ( g  +Q  h ) ) )
2422, 23eqtr3i 2485 . . . 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 9234 . . . . . . 7  |-  ( ( *Q `  ( g  +Q  h ) )  .Q  ( g  +Q  h ) )  =  ( ( g  +Q  h )  .Q  ( *Q `  ( g  +Q  h ) ) )
26 elprnq 9272 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
27 elprnq 9272 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  h  e.  Q. )
2826, 27anim12i 566 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( g  e.  Q.  /\  h  e.  Q. )
)
29 addclnq 9226 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
30 recidnq 9246 . . . . . . . 8  |-  ( ( g  +Q  h )  e.  Q.  ->  (
( g  +Q  h
)  .Q  ( *Q
`  ( g  +Q  h ) ) )  =  1Q )
3128, 29, 303syl 20 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( ( g  +Q  h )  .Q  ( *Q `  ( g  +Q  h ) ) )  =  1Q )
3225, 31syl5eq 2507 . . . . . 6  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( ( *Q `  ( g  +Q  h
) )  .Q  (
g  +Q  h ) )  =  1Q )
3332oveq2d 6217 . . . . 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 9244 . . . . 5  |-  ( x  e.  Q.  ->  (
x  .Q  1Q )  =  x )
3533, 34sylan9eq 2515 . . . 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 2507 . . 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 2523 . 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 214 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Q.cnq 9131   1Qc1q 9132    +Q cplq 9134    .Q cmq 9135   *Qcrq 9136    <Q cltq 9137   P.cnp 9138    +P. cpp 9140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-omul 7036  df-er 7212  df-ni 9153  df-pli 9154  df-mi 9155  df-lti 9156  df-plpq 9189  df-mpq 9190  df-ltpq 9191  df-enq 9192  df-nq 9193  df-erq 9194  df-plq 9195  df-mq 9196  df-1nq 9197  df-rq 9198  df-ltnq 9199  df-np 9262  df-plp 9264
This theorem is referenced by:  addclpr  9299
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