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Theorem addclprlem2 8636
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
Dummy variables  y 
z  w  v are mutually distinct and distinct from all other variables.
Allowed substitution hints:    A( g, h)    B( g, h)

Proof of Theorem addclprlem2
StepHypRef Expression
1 addclprlem1 8635 . . . . 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 697 . . . 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 8635 . . . . . 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 8570 . . . . . . 7  |-  ( g  +Q  h )  =  ( h  +Q  g
)
54breq2i 4032 . . . . . 6  |-  ( x 
<Q  ( g  +Q  h
)  <->  x  <Q  ( h  +Q  g ) )
64fveq2i 5488 . . . . . . . . 9  |-  ( *Q
`  ( g  +Q  h ) )  =  ( *Q `  (
h  +Q  g ) )
76oveq2i 5830 . . . . . . . 8  |-  ( x  .Q  ( *Q `  ( g  +Q  h
) ) )  =  ( x  .Q  ( *Q `  ( h  +Q  g ) ) )
87oveq1i 5829 . . . . . . 7  |-  ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  h )  =  ( ( x  .Q  ( *Q `  ( h  +Q  g ) ) )  .Q  h )
98eleq1i 2347 . . . . . 6  |-  ( ( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  h )  e.  B  <->  ( ( x  .Q  ( *Q `  ( h  +Q  g
) ) )  .Q  h )  e.  B
)
103, 5, 93imtr4g 263 . . . . 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 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  h )  e.  B
) )
122, 11jcad 521 . . 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 445 . . . 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 445 . . . . 5  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  A  e.  P. )
15 simpl 445 . . . . 5  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  B  e.  P. )
1614, 15anim12i 551 . . . 4  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( A  e.  P.  /\  B  e.  P. )
)
17 df-plp 8602 . . . . 5  |-  +P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  +Q  z ) } )
18 addclnq 8564 . . . . 5  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
1917, 18genpprecl 8620 . . . 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 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 8580 . . . . 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 8578 . . . . 5  |-  ( ( x  .Q  ( *Q
`  ( g  +Q  h ) ) )  .Q  ( g  +Q  h ) )  =  ( x  .Q  (
( *Q `  (
g  +Q  h ) )  .Q  ( g  +Q  h ) ) )
2422, 23eqtr3i 2306 . . . 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 8572 . . . . . . 7  |-  ( ( *Q `  ( g  +Q  h ) )  .Q  ( g  +Q  h ) )  =  ( ( g  +Q  h )  .Q  ( *Q `  ( g  +Q  h ) ) )
26 elprnq 8610 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  g  e.  A )  ->  g  e.  Q. )
27 elprnq 8610 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  h  e.  B )  ->  h  e.  Q. )
2826, 27anim12i 551 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( g  e.  Q.  /\  h  e.  Q. )
)
29 addclnq 8564 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
30 recidnq 8584 . . . . . . . 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 2328 . . . . . 6  |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e. 
P.  /\  h  e.  B ) )  -> 
( ( *Q `  ( g  +Q  h
) )  .Q  (
g  +Q  h ) )  =  1Q )
3332oveq2d 5835 . . . . 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 8582 . . . . 5  |-  ( x  e.  Q.  ->  (
x  .Q  1Q )  =  x )
3533, 34sylan9eq 2336 . . . 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 2328 . . 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 2350 . 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 207 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 6    /\ wa 360    = wceq 1624    e. wcel 1685   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Q.cnq 8469   1Qc1q 8470    +Q cplq 8472    .Q cmq 8473   *Qcrq 8474    <Q cltq 8475   P.cnp 8476    +P. cpp 8478
This theorem is referenced by:  addclpr  8637
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6655  df-ni 8491  df-pli 8492  df-mi 8493  df-lti 8494  df-plpq 8527  df-mpq 8528  df-ltpq 8529  df-enq 8530  df-nq 8531  df-erq 8532  df-plq 8533  df-mq 8534  df-1nq 8535  df-rq 8536  df-ltnq 8537  df-np 8600  df-plp 8602
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