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Theorem addclprlem2 8894
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 8893 . . . . 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 696 . . . 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 8893 . . . . . 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 8828 . . . . . . 7  |-  ( g  +Q  h )  =  ( h  +Q  g
)
54breq2i 4220 . . . . . 6  |-  ( x 
<Q  ( g  +Q  h
)  <->  x  <Q  ( h  +Q  g ) )
64fveq2i 5731 . . . . . . . . 9  |-  ( *Q
`  ( g  +Q  h ) )  =  ( *Q `  (
h  +Q  g ) )
76oveq2i 6092 . . . . . . . 8  |-  ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  =  ( x  .Q  ( *Q `  ( h  +Q  g ) ) )
87oveq1i 6091 . . . . . . 7  |-  ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  h )  =  ( ( x  .Q  ( *Q `  ( h  +Q  g ) ) )  .Q  h )
98eleq1i 2499 . . . . . 6  |-  ( ( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  h )  e.  B  <->  ( ( x  .Q  ( *Q `  ( h  +Q  g
) ) )  .Q  h )  e.  B
)
103, 5, 93imtr4g 262 . . . . 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 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  h )  e.  B
) )
122, 11jcad 520 . . 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 444 . . . 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 444 . . . . 5  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  A  e.  P. )
15 simpl 444 . . . . 5  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  B  e.  P. )
1614, 15anim12i 550 . . . 4  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( A  e.  P.  /\  B  e.  P. )
)
17 df-plp 8860 . . . . 5  |-  +P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  +Q  z ) } )
18 addclnq 8822 . . . . 5  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
1917, 18genpprecl 8878 . . . 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 19 . . 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 42 . 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 8838 . . . . 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 8836 . . . . 5  |-  ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  ( g  +Q  h ) )  =  ( x  .Q  (
( *Q `  (
g  +Q  h ) )  .Q  ( g  +Q  h ) ) )
2422, 23eqtr3i 2458 . . . 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 8830 . . . . . . 7  |-  ( ( *Q `  ( g  +Q  h ) )  .Q  ( g  +Q  h ) )  =  ( ( g  +Q  h )  .Q  ( *Q `  ( g  +Q  h ) ) )
26 elprnq 8868 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
27 elprnq 8868 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  h  e.  Q. )
2826, 27anim12i 550 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( g  e.  Q.  /\  h  e.  Q. )
)
29 addclnq 8822 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
30 recidnq 8842 . . . . . . . 8  |-  ( ( g  +Q  h )  e.  Q.  ->  (
( g  +Q  h
)  .Q  ( *Q
`  ( g  +Q  h ) ) )  =  1Q )
3128, 29, 303syl 19 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( ( g  +Q  h )  .Q  ( *Q `  ( g  +Q  h ) ) )  =  1Q )
3225, 31syl5eq 2480 . . . . . 6  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( ( *Q `  ( g  +Q  h
) )  .Q  (
g  +Q  h ) )  =  1Q )
3332oveq2d 6097 . . . . 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 8840 . . . . 5  |-  ( x  e.  Q.  ->  (
x  .Q  1Q )  =  x )
3533, 34sylan9eq 2488 . . . 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 2480 . . 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 2502 . 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 206 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 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Q.cnq 8727   1Qc1q 8728    +Q cplq 8730    .Q cmq 8731   *Qcrq 8732    <Q cltq 8733   P.cnp 8734    +P. cpp 8736
This theorem is referenced by:  addclpr  8895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-ni 8749  df-pli 8750  df-mi 8751  df-lti 8752  df-plpq 8785  df-mpq 8786  df-ltpq 8787  df-enq 8788  df-nq 8789  df-erq 8790  df-plq 8791  df-mq 8792  df-1nq 8793  df-rq 8794  df-ltnq 8795  df-np 8858  df-plp 8860
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