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Theorem addclpr 7357
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 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)

Proof of Theorem addclpr
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑔 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iplp 7288 . . . 4 +P = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦 +Q 𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦 +Q 𝑧))}⟩)
21genpelxp 7331 . . 3 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3 addclnq 7195 . . . 4 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
41, 3genpml 7337 . . 3 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))
51, 3genpmu 7338 . . 3 ((𝐴P𝐵P) → ∃𝑟Q 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))
62, 4, 5jca32 308 . 2 ((𝐴P𝐵P) → ((𝐴 +P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ ∃𝑟Q 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
7 ltanqg 7220 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧 +Q 𝑥) <Q (𝑧 +Q 𝑦)))
8 addcomnqg 7201 . . . . 5 ((𝑥Q𝑦Q) → (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥))
9 addnqprl 7349 . . . . 5 ((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → 𝑥 ∈ (1st ‘(𝐴 +P 𝐵))))
101, 3, 7, 8, 9genprndl 7341 . . . 4 ((𝐴P𝐵P) → ∀𝑞Q (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴 +P 𝐵)))))
11 addnqpru 7350 . . . . 5 ((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑔 +Q ) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴 +P 𝐵))))
121, 3, 7, 8, 11genprndu 7342 . . . 4 ((𝐴P𝐵P) → ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴 +P 𝐵)))))
1310, 12jca 304 . . 3 ((𝐴P𝐵P) → (∀𝑞Q (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴 +P 𝐵)))) ∧ ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))))))
141, 3, 7, 8genpdisj 7343 . . 3 ((𝐴P𝐵P) → ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))))
15 addlocpr 7356 . . 3 ((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
1613, 14, 153jca 1161 . 2 ((𝐴P𝐵P) → ((∀𝑞Q (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴 +P 𝐵)))) ∧ ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵))))))
17 elnp1st2nd 7296 . 2 ((𝐴 +P 𝐵) ∈ P ↔ (((𝐴 +P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ ∃𝑟Q 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))) ∧ ((∀𝑞Q (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴 +P 𝐵)))) ∧ ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))))
186, 16, 17sylanbrc 413 1 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  w3a 962  wcel 1480  wral 2416  wrex 2417  𝒫 cpw 3510   class class class wbr 3929   × cxp 4537  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7100   +Q cplq 7102   <Q cltq 7105  Pcnp 7111   +P cpp 7113
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-iplp 7288
This theorem is referenced by:  addnqprlemfl  7379  addnqprlemfu  7380  addnqpr  7381  addassprg  7399  distrlem1prl  7402  distrlem1pru  7403  distrlem4prl  7404  distrlem4pru  7405  distrprg  7408  ltaddpr  7417  ltexpri  7433  addcanprleml  7434  addcanprlemu  7435  ltaprlem  7438  ltaprg  7439  prplnqu  7440  addextpr  7441  caucvgprlemcanl  7464  cauappcvgprlemladdru  7476  cauappcvgprlemladdrl  7477  cauappcvgprlemladd  7478  cauappcvgprlem1  7479  caucvgprlemladdrl  7498  caucvgprlem1  7499  caucvgprprlemnbj  7513  caucvgprprlemopu  7519  caucvgprprlemloc  7523  caucvgprprlemexbt  7526  caucvgprprlemexb  7527  caucvgprprlemaddq  7528  caucvgprprlem2  7530  enrer  7555  addcmpblnr  7559  mulcmpblnrlemg  7560  mulcmpblnr  7561  ltsrprg  7567  1sr  7571  m1r  7572  addclsr  7573  mulclsr  7574  addasssrg  7576  mulasssrg  7578  distrsrg  7579  m1p1sr  7580  m1m1sr  7581  lttrsr  7582  ltsosr  7584  0lt1sr  7585  0idsr  7587  1idsr  7588  00sr  7589  ltasrg  7590  recexgt0sr  7593  mulgt0sr  7598  aptisr  7599  mulextsr1lem  7600  mulextsr1  7601  archsr  7602  srpospr  7603  prsrcl  7604  prsradd  7606  prsrlt  7607  caucvgsrlemcau  7613  caucvgsrlemgt1  7615  mappsrprg  7624  map2psrprg  7625  pitonnlem1p1  7666  pitonnlem2  7667  pitonn  7668  pitoregt0  7669  pitore  7670  recnnre  7671  recidpirqlemcalc  7677  recidpirq  7678
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