Theorem addclpr 7724

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 7655	. . . 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 7698	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. ( ~P Q. X. ~P Q. ) )
3		addclnq 7562	. . . 4 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) e. Q. )
4	1, 3	genpml 7704	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A +P. B ) ) )
5	1, 3	genpmu 7705	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. r e. Q. r e. ( 2nd ` ( A +P. B ) ) )
6	2, 4, 5	jca32 310	. 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 7587	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z +Q x ) <Q ( z +Q y ) ) )
8		addcomnqg 7568	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( x +Q y ) = ( y +Q x ) )
9		addnqprl 7716	. . . . 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 7708	. . . 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 7717	. . . . 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 7709	. . . 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 306	. . 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 7710	. . 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 7723	. . 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 1201	. 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 7663	. 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 417	1 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   e. wcel 2200   A.wral 2508   E.wrex 2509   ~Pcpw 3649   class class class wbr 4083   X. cxp 4717   ` cfv 5318  (class class class)co 6001   1stc1st 6284   2ndc2nd 6285   Q.cnq 7467   +Q cplq 7469   <Q cltq 7472   P.cnp 7478   +P. cpp 7480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-2o 6563  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-pli 7492  df-mi 7493  df-lti 7494  df-plpq 7531  df-mpq 7532  df-enq 7534  df-nqqs 7535  df-plqqs 7536  df-mqqs 7537  df-1nqqs 7538  df-rq 7539  df-ltnqqs 7540  df-enq0 7611  df-nq0 7612  df-0nq0 7613  df-plq0 7614  df-mq0 7615  df-inp 7653  df-iplp 7655

This theorem is referenced by:  addnqprlemfl  7746  addnqprlemfu  7747  addnqpr  7748  addassprg  7766  distrlem1prl  7769  distrlem1pru  7770  distrlem4prl  7771  distrlem4pru  7772  distrprg  7775  ltaddpr  7784  ltexpri  7800  addcanprleml  7801  addcanprlemu  7802  ltaprlem  7805  ltaprg  7806  prplnqu  7807  addextpr  7808  caucvgprlemcanl  7831  cauappcvgprlemladdru  7843  cauappcvgprlemladdrl  7844  cauappcvgprlemladd  7845  cauappcvgprlem1  7846  caucvgprlemladdrl  7865  caucvgprlem1  7866  caucvgprprlemnbj  7880  caucvgprprlemopu  7886  caucvgprprlemloc  7890  caucvgprprlemexbt  7893  caucvgprprlemexb  7894  caucvgprprlemaddq  7895  caucvgprprlem2  7897  enrer  7922  addcmpblnr  7926  mulcmpblnrlemg  7927  mulcmpblnr  7928  ltsrprg  7934  1sr  7938  m1r  7939  addclsr  7940  mulclsr  7941  addasssrg  7943  mulasssrg  7945  distrsrg  7946  m1p1sr  7947  m1m1sr  7948  lttrsr  7949  ltsosr  7951  0lt1sr  7952  0idsr  7954  1idsr  7955  00sr  7956  ltasrg  7957  recexgt0sr  7960  mulgt0sr  7965  aptisr  7966  mulextsr1lem  7967  mulextsr1  7968  archsr  7969  srpospr  7970  prsrcl  7971  prsradd  7973  prsrlt  7974  caucvgsrlemcau  7980  caucvgsrlemgt1  7982  mappsrprg  7991  map2psrprg  7992  pitonnlem1p1  8033  pitonnlem2  8034  pitonn  8035  pitoregt0  8036  pitore  8037  recnnre  8038  recidpirqlemcalc  8044  recidpirq  8045