Theorem addclpr 7043

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 6974	. . . 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 7017	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. ( ~P Q. X. ~P Q. ) )
3		addclnq 6881	. . . 4 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) e. Q. )
4	1, 3	genpml 7023	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A +P. B ) ) )
5	1, 3	genpmu 7024	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. r e. Q. r e. ( 2nd ` ( A +P. B ) ) )
6	2, 4, 5	jca32 303	. 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 6906	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z +Q x ) <Q ( z +Q y ) ) )
8		addcomnqg 6887	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( x +Q y ) = ( y +Q x ) )
9		addnqprl 7035	. . . . 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 7027	. . . 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 7036	. . . . 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 7028	. . . 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 300	. . 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 7029	. . 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 7042	. . 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 1121	. 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 6982	. 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 408	1 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 662   /\ w3a 922   e. wcel 1436   A.wral 2355   E.wrex 2356   ~Pcpw 3415   class class class wbr 3822   X. cxp 4411   ` cfv 4983  (class class class)co 5615   1stc1st 5868   2ndc2nd 5869   Q.cnq 6786   +Q cplq 6788   <Q cltq 6791   P.cnp 6797   +P. cpp 6799

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378

This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-eprel 4092  df-id 4096  df-po 4099  df-iso 4100  df-iord 4169  df-on 4171  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-irdg 6091  df-1o 6137  df-2o 6138  df-oadd 6141  df-omul 6142  df-er 6246  df-ec 6248  df-qs 6252  df-ni 6810  df-pli 6811  df-mi 6812  df-lti 6813  df-plpq 6850  df-mpq 6851  df-enq 6853  df-nqqs 6854  df-plqqs 6855  df-mqqs 6856  df-1nqqs 6857  df-rq 6858  df-ltnqqs 6859  df-enq0 6930  df-nq0 6931  df-0nq0 6932  df-plq0 6933  df-mq0 6934  df-inp 6972  df-iplp 6974

This theorem is referenced by:  addnqprlemfl  7065  addnqprlemfu  7066  addnqpr  7067  addassprg  7085  distrlem1prl  7088  distrlem1pru  7089  distrlem4prl  7090  distrlem4pru  7091  distrprg  7094  ltaddpr  7103  ltexpri  7119  addcanprleml  7120  addcanprlemu  7121  ltaprlem  7124  ltaprg  7125  prplnqu  7126  addextpr  7127  caucvgprlemcanl  7150  cauappcvgprlemladdru  7162  cauappcvgprlemladdrl  7163  cauappcvgprlemladd  7164  cauappcvgprlem1  7165  caucvgprlemladdrl  7184  caucvgprlem1  7185  caucvgprprlemnbj  7199  caucvgprprlemopu  7205  caucvgprprlemloc  7209  caucvgprprlemexbt  7212  caucvgprprlemexb  7213  caucvgprprlemaddq  7214  caucvgprprlem2  7216  enrer  7228  addcmpblnr  7232  mulcmpblnrlemg  7233  mulcmpblnr  7234  ltsrprg  7240  1sr  7244  m1r  7245  addclsr  7246  mulclsr  7247  addasssrg  7249  mulasssrg  7251  distrsrg  7252  m1p1sr  7253  m1m1sr  7254  lttrsr  7255  ltsosr  7257  0lt1sr  7258  0idsr  7260  1idsr  7261  00sr  7262  ltasrg  7263  recexgt0sr  7266  mulgt0sr  7270  aptisr  7271  mulextsr1lem  7272  mulextsr1  7273  archsr  7274  srpospr  7275  prsrcl  7276  prsradd  7278  prsrlt  7279  caucvgsrlemcau  7285  caucvgsrlemgt1  7287  pitonnlem1p1  7330  pitonnlem2  7331  pitonn  7332  pitoregt0  7333  pitore  7334  recnnre  7335  recidpirqlemcalc  7341  recidpirq  7342