Theorem addclpr 7469

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.

Step	Hyp	Ref	Expression
1		df-iplp 7400	. . . 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 ) ) } >. )
2	1	genpelxp 7443	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. ( ~P Q. X. ~P Q. ) )
3		addclnq 7307	. . . 4 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) e. Q. )
4	1, 3	genpml 7449	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A +P. B ) ) )
5	1, 3	genpmu 7450	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. r e. Q. r e. ( 2nd ` ( A +P. B ) ) )
6	2, 4, 5	jca32 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 7332	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z +Q x ) <Q ( z +Q y ) ) )
8		addcomnqg 7313	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( x +Q y ) = ( y +Q x ) )
9		addnqprl 7461	. . . . 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 ) ) ) )
10	1, 3, 7, 8, 9	genprndl 7453	. . . 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 7462	. . . . 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 ) ) ) )
12	1, 3, 7, 8, 11	genprndu 7454	. . . 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 ) ) ) ) )
13	10, 12	jca 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 ) ) ) ) ) )
14	1, 3, 7, 8	genpdisj 7455	. . 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 7468	. . 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 ) ) ) ) )
16	13, 14, 15	3jca 1166	. 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 7408	. 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 ) ) ) ) ) ) )
18	6, 16, 17	sylanbrc 414	1 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 967   e. wcel 2135   A.wral 2442   E.wrex 2443   ~Pcpw 3553   class class class wbr 3976   X. cxp 4596   ` cfv 5182  (class class class)co 5836   1stc1st 6098   2ndc2nd 6099   Q.cnq 7212   +Q cplq 7214   <Q cltq 7217   P.cnp 7223   +P. cpp 7225

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-eprel 4261  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-irdg 6329  df-1o 6375  df-2o 6376  df-oadd 6379  df-omul 6380  df-er 6492  df-ec 6494  df-qs 6498  df-ni 7236  df-pli 7237  df-mi 7238  df-lti 7239  df-plpq 7276  df-mpq 7277  df-enq 7279  df-nqqs 7280  df-plqqs 7281  df-mqqs 7282  df-1nqqs 7283  df-rq 7284  df-ltnqqs 7285  df-enq0 7356  df-nq0 7357  df-0nq0 7358  df-plq0 7359  df-mq0 7360  df-inp 7398  df-iplp 7400

This theorem is referenced by:  addnqprlemfl  7491  addnqprlemfu  7492  addnqpr  7493  addassprg  7511  distrlem1prl  7514  distrlem1pru  7515  distrlem4prl  7516  distrlem4pru  7517  distrprg  7520  ltaddpr  7529  ltexpri  7545  addcanprleml  7546  addcanprlemu  7547  ltaprlem  7550  ltaprg  7551  prplnqu  7552  addextpr  7553  caucvgprlemcanl  7576  cauappcvgprlemladdru  7588  cauappcvgprlemladdrl  7589  cauappcvgprlemladd  7590  cauappcvgprlem1  7591  caucvgprlemladdrl  7610  caucvgprlem1  7611  caucvgprprlemnbj  7625  caucvgprprlemopu  7631  caucvgprprlemloc  7635  caucvgprprlemexbt  7638  caucvgprprlemexb  7639  caucvgprprlemaddq  7640  caucvgprprlem2  7642  enrer  7667  addcmpblnr  7671  mulcmpblnrlemg  7672  mulcmpblnr  7673  ltsrprg  7679  1sr  7683  m1r  7684  addclsr  7685  mulclsr  7686  addasssrg  7688  mulasssrg  7690  distrsrg  7691  m1p1sr  7692  m1m1sr  7693  lttrsr  7694  ltsosr  7696  0lt1sr  7697  0idsr  7699  1idsr  7700  00sr  7701  ltasrg  7702  recexgt0sr  7705  mulgt0sr  7710  aptisr  7711  mulextsr1lem  7712  mulextsr1  7713  archsr  7714  srpospr  7715  prsrcl  7716  prsradd  7718  prsrlt  7719  caucvgsrlemcau  7725  caucvgsrlemgt1  7727  mappsrprg  7736  map2psrprg  7737  pitonnlem1p1  7778  pitonnlem2  7779  pitonn  7780  pitoregt0  7781  pitore  7782  recnnre  7783  recidpirqlemcalc  7789  recidpirq  7790