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Theorem addclpr 7364
Description: Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. Combination of Lemma 11.13 and Lemma 11.16 in [BauerTaylor], p. 53. (Contributed by NM, 13-Mar-1996.)
Assertion
Ref Expression
addclpr  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )

Proof of Theorem addclpr
Dummy variables  x  y  z  w  v  g  h  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iplp 7295 . . . 4  |-  +P.  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y  +Q  z
) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y  +Q  z
) ) } >. )
21genpelxp 7338 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  ( ~P Q.  X.  ~P Q. ) )
3 addclnq 7202 . . . 4  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
41, 3genpml 7344 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( A  +P.  B
) ) )
51, 3genpmu 7345 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. r  e.  Q.  r  e.  ( 2nd `  ( A  +P.  B
) ) )
62, 4, 5jca32 308 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A  +P.  B )  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  ( A  +P.  B ) )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
7 ltanqg 7227 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z  +Q  x )  <Q  (
z  +Q  y ) ) )
8 addcomnqg 7208 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  +Q  y
)  =  ( y  +Q  x ) )
9 addnqprl 7356 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  g  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 1st `  B
) ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g  +Q  h )  ->  x  e.  ( 1st `  ( A  +P.  B ) ) ) )
101, 3, 7, 8, 9genprndl 7348 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. q  e.  Q.  ( q  e.  ( 1st `  ( A  +P.  B ) )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  ( A  +P.  B ) ) ) ) )
11 addnqpru 7357 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  g  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 2nd `  B
) ) )  /\  x  e.  Q. )  ->  ( ( g  +Q  h )  <Q  x  ->  x  e.  ( 2nd `  ( A  +P.  B
) ) ) )
121, 3, 7, 8, 11genprndu 7349 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. r  e.  Q.  ( r  e.  ( 2nd `  ( A  +P.  B ) )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
1310, 12jca 304 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A. q  e. 
Q.  ( q  e.  ( 1st `  ( A  +P.  B ) )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  ( A  +P.  B ) ) ) )  /\  A. r  e. 
Q.  ( r  e.  ( 2nd `  ( A  +P.  B ) )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A  +P.  B ) ) ) ) ) )
141, 3, 7, 8genpdisj 7350 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  ( A  +P.  B ) )  /\  q  e.  ( 2nd `  ( A  +P.  B ) ) ) )
15 addlocpr 7363 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. q  e.  Q.  A. r  e.  Q.  (
q  <Q  r  ->  (
q  e.  ( 1st `  ( A  +P.  B
) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
1613, 14, 153jca 1161 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( A. q  e.  Q.  ( q  e.  ( 1st `  ( A  +P.  B ) )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  ( A  +P.  B ) ) ) )  /\  A. r  e. 
Q.  ( r  e.  ( 2nd `  ( A  +P.  B ) )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )  /\  A. q  e.  Q.  -.  ( q  e.  ( 1st `  ( A  +P.  B ) )  /\  q  e.  ( 2nd `  ( A  +P.  B ) ) )  /\  A. q  e.  Q.  A. r  e. 
Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  ( A  +P.  B ) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) ) ) )
17 elnp1st2nd 7303 . 2  |-  ( ( A  +P.  B )  e.  P.  <->  ( (
( A  +P.  B
)  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  ( A  +P.  B ) )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  ( A  +P.  B ) ) ) )  /\  (
( A. q  e. 
Q.  ( q  e.  ( 1st `  ( A  +P.  B ) )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  ( A  +P.  B ) ) ) )  /\  A. r  e. 
Q.  ( r  e.  ( 2nd `  ( A  +P.  B ) )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )  /\  A. q  e.  Q.  -.  ( q  e.  ( 1st `  ( A  +P.  B ) )  /\  q  e.  ( 2nd `  ( A  +P.  B ) ) )  /\  A. q  e.  Q.  A. r  e. 
Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  ( A  +P.  B ) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) ) ) ) )
186, 16, 17sylanbrc 413 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    e. wcel 1480   A.wral 2416   E.wrex 2417   ~Pcpw 3510   class class class wbr 3932    X. cxp 4540   ` cfv 5126  (class class class)co 5777   1stc1st 6039   2ndc2nd 6040   Q.cnq 7107    +Q cplq 7109    <Q cltq 7112   P.cnp 7118    +P. cpp 7120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-iinf 4505
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-eprel 4214  df-id 4218  df-po 4221  df-iso 4222  df-iord 4291  df-on 4293  df-suc 4296  df-iom 4508  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-recs 6205  df-irdg 6270  df-1o 6316  df-2o 6317  df-oadd 6320  df-omul 6321  df-er 6432  df-ec 6434  df-qs 6438  df-ni 7131  df-pli 7132  df-mi 7133  df-lti 7134  df-plpq 7171  df-mpq 7172  df-enq 7174  df-nqqs 7175  df-plqqs 7176  df-mqqs 7177  df-1nqqs 7178  df-rq 7179  df-ltnqqs 7180  df-enq0 7251  df-nq0 7252  df-0nq0 7253  df-plq0 7254  df-mq0 7255  df-inp 7293  df-iplp 7295
This theorem is referenced by:  addnqprlemfl  7386  addnqprlemfu  7387  addnqpr  7388  addassprg  7406  distrlem1prl  7409  distrlem1pru  7410  distrlem4prl  7411  distrlem4pru  7412  distrprg  7415  ltaddpr  7424  ltexpri  7440  addcanprleml  7441  addcanprlemu  7442  ltaprlem  7445  ltaprg  7446  prplnqu  7447  addextpr  7448  caucvgprlemcanl  7471  cauappcvgprlemladdru  7483  cauappcvgprlemladdrl  7484  cauappcvgprlemladd  7485  cauappcvgprlem1  7486  caucvgprlemladdrl  7505  caucvgprlem1  7506  caucvgprprlemnbj  7520  caucvgprprlemopu  7526  caucvgprprlemloc  7530  caucvgprprlemexbt  7533  caucvgprprlemexb  7534  caucvgprprlemaddq  7535  caucvgprprlem2  7537  enrer  7562  addcmpblnr  7566  mulcmpblnrlemg  7567  mulcmpblnr  7568  ltsrprg  7574  1sr  7578  m1r  7579  addclsr  7580  mulclsr  7581  addasssrg  7583  mulasssrg  7585  distrsrg  7586  m1p1sr  7587  m1m1sr  7588  lttrsr  7589  ltsosr  7591  0lt1sr  7592  0idsr  7594  1idsr  7595  00sr  7596  ltasrg  7597  recexgt0sr  7600  mulgt0sr  7605  aptisr  7606  mulextsr1lem  7607  mulextsr1  7608  archsr  7609  srpospr  7610  prsrcl  7611  prsradd  7613  prsrlt  7614  caucvgsrlemcau  7620  caucvgsrlemgt1  7622  mappsrprg  7631  map2psrprg  7632  pitonnlem1p1  7673  pitonnlem2  7674  pitonn  7675  pitoregt0  7676  pitore  7677  recnnre  7678  recidpirqlemcalc  7684  recidpirq  7685
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