Theorem addclpr 7499

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 7430	. . . 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 7473	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. ( ~P Q. X. ~P Q. ) )
3		addclnq 7337	. . . 4 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) e. Q. )
4	1, 3	genpml 7479	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A +P. B ) ) )
5	1, 3	genpmu 7480	. . 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 7362	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z +Q x ) <Q ( z +Q y ) ) )
8		addcomnqg 7343	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( x +Q y ) = ( y +Q x ) )
9		addnqprl 7491	. . . . 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 7483	. . . 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 7492	. . . . 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 7484	. . . 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 7485	. . 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 7498	. . 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 1172	. 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 7438	. 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 415	1 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   /\ w3a 973   e. wcel 2141   A.wral 2448   E.wrex 2449   ~Pcpw 3566   class class class wbr 3989   X. cxp 4609   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   +Q cplq 7244   <Q cltq 7247   P.cnp 7253   +P. cpp 7255

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430

This theorem is referenced by:  addnqprlemfl  7521  addnqprlemfu  7522  addnqpr  7523  addassprg  7541  distrlem1prl  7544  distrlem1pru  7545  distrlem4prl  7546  distrlem4pru  7547  distrprg  7550  ltaddpr  7559  ltexpri  7575  addcanprleml  7576  addcanprlemu  7577  ltaprlem  7580  ltaprg  7581  prplnqu  7582  addextpr  7583  caucvgprlemcanl  7606  cauappcvgprlemladdru  7618  cauappcvgprlemladdrl  7619  cauappcvgprlemladd  7620  cauappcvgprlem1  7621  caucvgprlemladdrl  7640  caucvgprlem1  7641  caucvgprprlemnbj  7655  caucvgprprlemopu  7661  caucvgprprlemloc  7665  caucvgprprlemexbt  7668  caucvgprprlemexb  7669  caucvgprprlemaddq  7670  caucvgprprlem2  7672  enrer  7697  addcmpblnr  7701  mulcmpblnrlemg  7702  mulcmpblnr  7703  ltsrprg  7709  1sr  7713  m1r  7714  addclsr  7715  mulclsr  7716  addasssrg  7718  mulasssrg  7720  distrsrg  7721  m1p1sr  7722  m1m1sr  7723  lttrsr  7724  ltsosr  7726  0lt1sr  7727  0idsr  7729  1idsr  7730  00sr  7731  ltasrg  7732  recexgt0sr  7735  mulgt0sr  7740  aptisr  7741  mulextsr1lem  7742  mulextsr1  7743  archsr  7744  srpospr  7745  prsrcl  7746  prsradd  7748  prsrlt  7749  caucvgsrlemcau  7755  caucvgsrlemgt1  7757  mappsrprg  7766  map2psrprg  7767  pitonnlem1p1  7808  pitonnlem2  7809  pitonn  7810  pitoregt0  7811  pitore  7812  recnnre  7813  recidpirqlemcalc  7819  recidpirq  7820