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Theorem addclprlem2 9384
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 9383 . . . . 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 712 . . . 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 9383 . . . . . 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 9318 . . . . . . 7  |-  ( g  +Q  h )  =  ( h  +Q  g
)
54breq2i 4447 . . . . . 6  |-  ( x 
<Q  ( g  +Q  h
)  <->  x  <Q  ( h  +Q  g ) )
64fveq2i 5851 . . . . . . . . 9  |-  ( *Q
`  ( g  +Q  h ) )  =  ( *Q `  (
h  +Q  g ) )
76oveq2i 6281 . . . . . . . 8  |-  ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  =  ( x  .Q  ( *Q `  ( h  +Q  g ) ) )
87oveq1i 6280 . . . . . . 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 711 . . . 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 531 . . 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 455 . . . 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 455 . . . . 5  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  A  e.  P. )
15 simpl 455 . . . . 5  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  B  e.  P. )
1614, 15anim12i 564 . . . 4  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( A  e.  P.  /\  B  e.  P. )
)
17 df-plp 9350 . . . . 5  |-  +P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  +Q  z ) } )
18 addclnq 9312 . . . . 5  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
1917, 18genpprecl 9368 . . . 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 9328 . . . . 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 9326 . . . . 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 9320 . . . . . . 7  |-  ( ( *Q `  ( g  +Q  h ) )  .Q  ( g  +Q  h ) )  =  ( ( g  +Q  h )  .Q  ( *Q `  ( g  +Q  h ) ) )
26 elprnq 9358 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
27 elprnq 9358 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  h  e.  Q. )
2826, 27anim12i 564 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( g  e.  Q.  /\  h  e.  Q. )
)
29 addclnq 9312 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
30 recidnq 9332 . . . . . . . 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 6286 . . . . 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 9330 . . . . 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 367    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Q.cnq 9219   1Qc1q 9220    +Q cplq 9222    .Q cmq 9223   *Qcrq 9224    <Q cltq 9225   P.cnp 9226    +P. cpp 9228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-er 7303  df-ni 9239  df-pli 9240  df-mi 9241  df-lti 9242  df-plpq 9275  df-mpq 9276  df-ltpq 9277  df-enq 9278  df-nq 9279  df-erq 9280  df-plq 9281  df-mq 9282  df-1nq 9283  df-rq 9284  df-ltnq 9285  df-np 9348  df-plp 9350
This theorem is referenced by:  addclpr  9385
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