Theorem addclpr 7536

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 7467	. . . 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 7510	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. ( ~P Q. X. ~P Q. ) )
3		addclnq 7374	. . . 4 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) e. Q. )
4	1, 3	genpml 7516	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A +P. B ) ) )
5	1, 3	genpmu 7517	. . 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 7399	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z +Q x ) <Q ( z +Q y ) ) )
8		addcomnqg 7380	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( x +Q y ) = ( y +Q x ) )
9		addnqprl 7528	. . . . 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 7520	. . . 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 7529	. . . . 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 7521	. . . 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 7522	. . 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 7535	. . 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 1177	. 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 7475	. 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 708   /\ w3a 978   e. wcel 2148   A.wral 2455   E.wrex 2456   ~Pcpw 3576   class class class wbr 4004   X. cxp 4625   ` cfv 5217  (class class class)co 5875   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279   +Q cplq 7281   <Q cltq 7284   P.cnp 7290   +P. cpp 7292

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-iplp 7467

This theorem is referenced by:  addnqprlemfl  7558  addnqprlemfu  7559  addnqpr  7560  addassprg  7578  distrlem1prl  7581  distrlem1pru  7582  distrlem4prl  7583  distrlem4pru  7584  distrprg  7587  ltaddpr  7596  ltexpri  7612  addcanprleml  7613  addcanprlemu  7614  ltaprlem  7617  ltaprg  7618  prplnqu  7619  addextpr  7620  caucvgprlemcanl  7643  cauappcvgprlemladdru  7655  cauappcvgprlemladdrl  7656  cauappcvgprlemladd  7657  cauappcvgprlem1  7658  caucvgprlemladdrl  7677  caucvgprlem1  7678  caucvgprprlemnbj  7692  caucvgprprlemopu  7698  caucvgprprlemloc  7702  caucvgprprlemexbt  7705  caucvgprprlemexb  7706  caucvgprprlemaddq  7707  caucvgprprlem2  7709  enrer  7734  addcmpblnr  7738  mulcmpblnrlemg  7739  mulcmpblnr  7740  ltsrprg  7746  1sr  7750  m1r  7751  addclsr  7752  mulclsr  7753  addasssrg  7755  mulasssrg  7757  distrsrg  7758  m1p1sr  7759  m1m1sr  7760  lttrsr  7761  ltsosr  7763  0lt1sr  7764  0idsr  7766  1idsr  7767  00sr  7768  ltasrg  7769  recexgt0sr  7772  mulgt0sr  7777  aptisr  7778  mulextsr1lem  7779  mulextsr1  7780  archsr  7781  srpospr  7782  prsrcl  7783  prsradd  7785  prsrlt  7786  caucvgsrlemcau  7792  caucvgsrlemgt1  7794  mappsrprg  7803  map2psrprg  7804  pitonnlem1p1  7845  pitonnlem2  7846  pitonn  7847  pitoregt0  7848  pitore  7849  recnnre  7850  recidpirqlemcalc  7856  recidpirq  7857