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Theorem addclpr 6693
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 6624 . . . 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 6667 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  ( ~P Q.  X.  ~P Q. ) )
3 addclnq 6531 . . . 4  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
41, 3genpml 6673 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( A  +P.  B
) ) )
51, 3genpmu 6674 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. r  e.  Q.  r  e.  ( 2nd `  ( A  +P.  B
) ) )
62, 4, 5jca32 297 . 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 6556 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z  +Q  x )  <Q  (
z  +Q  y ) ) )
8 addcomnqg 6537 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  +Q  y
)  =  ( y  +Q  x ) )
9 addnqprl 6685 . . . . 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 6677 . . . 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 6686 . . . . 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 6678 . . . 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 294 . . 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 6679 . . 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 6692 . . 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 1095 . 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 6632 . 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 402 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 101    <-> wb 102    \/ wo 639    /\ w3a 896    e. wcel 1409   A.wral 2323   E.wrex 2324   ~Pcpw 3387   class class class wbr 3792    X. cxp 4371   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    +Q cplq 6438    <Q cltq 6441   P.cnp 6447    +P. cpp 6449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624
This theorem is referenced by:  addnqprlemfl  6715  addnqprlemfu  6716  addnqpr  6717  addassprg  6735  distrlem1prl  6738  distrlem1pru  6739  distrlem4prl  6740  distrlem4pru  6741  distrprg  6744  ltaddpr  6753  ltexpri  6769  addcanprleml  6770  addcanprlemu  6771  ltaprlem  6774  ltaprg  6775  prplnqu  6776  addextpr  6777  caucvgprlemcanl  6800  cauappcvgprlemladdru  6812  cauappcvgprlemladdrl  6813  cauappcvgprlemladd  6814  cauappcvgprlem1  6815  caucvgprlemladdrl  6834  caucvgprlem1  6835  caucvgprprlemnbj  6849  caucvgprprlemopu  6855  caucvgprprlemloc  6859  caucvgprprlemexbt  6862  caucvgprprlemexb  6863  caucvgprprlemaddq  6864  caucvgprprlem2  6866  enrer  6878  addcmpblnr  6882  mulcmpblnrlemg  6883  mulcmpblnr  6884  ltsrprg  6890  1sr  6894  m1r  6895  addclsr  6896  mulclsr  6897  addasssrg  6899  mulasssrg  6901  distrsrg  6902  m1p1sr  6903  m1m1sr  6904  lttrsr  6905  ltsosr  6907  0lt1sr  6908  0idsr  6910  1idsr  6911  00sr  6912  ltasrg  6913  recexgt0sr  6916  mulgt0sr  6920  aptisr  6921  mulextsr1lem  6922  mulextsr1  6923  archsr  6924  srpospr  6925  prsrcl  6926  prsradd  6928  prsrlt  6929  caucvgsrlemcau  6935  caucvgsrlemgt1  6937  pitonnlem1p1  6980  pitonnlem2  6981  pitonn  6982  pitoregt0  6983  pitore  6984  recnnre  6985  recidpirqlemcalc  6991  recidpirq  6992
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