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Theorem addclpr 7470
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 7401 . . . 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 7444 . . 3 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3 addclnq 7308 . . . 4 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
41, 3genpml 7450 . . 3 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))
51, 3genpmu 7451 . . 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 7333 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧 +Q 𝑥) <Q (𝑧 +Q 𝑦)))
8 addcomnqg 7314 . . . . 5 ((𝑥Q𝑦Q) → (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥))
9 addnqprl 7462 . . . . 5 ((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → 𝑥 ∈ (1st ‘(𝐴 +P 𝐵))))
101, 3, 7, 8, 9genprndl 7454 . . . 4 ((𝐴P𝐵P) → ∀𝑞Q (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴 +P 𝐵)))))
11 addnqpru 7463 . . . . 5 ((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑔 +Q ) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴 +P 𝐵))))
121, 3, 7, 8, 11genprndu 7455 . . . 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 7456 . . 3 ((𝐴P𝐵P) → ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))))
15 addlocpr 7469 . . 3 ((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
1613, 14, 153jca 1166 . 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 7409 . 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 414 1 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  w3a 967  wcel 2135  wral 2442  wrex 2443  𝒫 cpw 3554   class class class wbr 3977   × cxp 4597  cfv 5183  (class class class)co 5837  1st c1st 6099  2nd c2nd 6100  Qcnq 7213   +Q cplq 7215   <Q cltq 7218  Pcnp 7224   +P cpp 7226
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4092  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-iinf 4560
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2724  df-sbc 2948  df-csb 3042  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-int 3820  df-iun 3863  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-eprel 4262  df-id 4266  df-po 4269  df-iso 4270  df-iord 4339  df-on 4341  df-suc 4344  df-iom 4563  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190  df-fv 5191  df-ov 5840  df-oprab 5841  df-mpo 5842  df-1st 6101  df-2nd 6102  df-recs 6265  df-irdg 6330  df-1o 6376  df-2o 6377  df-oadd 6380  df-omul 6381  df-er 6493  df-ec 6495  df-qs 6499  df-ni 7237  df-pli 7238  df-mi 7239  df-lti 7240  df-plpq 7277  df-mpq 7278  df-enq 7280  df-nqqs 7281  df-plqqs 7282  df-mqqs 7283  df-1nqqs 7284  df-rq 7285  df-ltnqqs 7286  df-enq0 7357  df-nq0 7358  df-0nq0 7359  df-plq0 7360  df-mq0 7361  df-inp 7399  df-iplp 7401
This theorem is referenced by:  addnqprlemfl  7492  addnqprlemfu  7493  addnqpr  7494  addassprg  7512  distrlem1prl  7515  distrlem1pru  7516  distrlem4prl  7517  distrlem4pru  7518  distrprg  7521  ltaddpr  7530  ltexpri  7546  addcanprleml  7547  addcanprlemu  7548  ltaprlem  7551  ltaprg  7552  prplnqu  7553  addextpr  7554  caucvgprlemcanl  7577  cauappcvgprlemladdru  7589  cauappcvgprlemladdrl  7590  cauappcvgprlemladd  7591  cauappcvgprlem1  7592  caucvgprlemladdrl  7611  caucvgprlem1  7612  caucvgprprlemnbj  7626  caucvgprprlemopu  7632  caucvgprprlemloc  7636  caucvgprprlemexbt  7639  caucvgprprlemexb  7640  caucvgprprlemaddq  7641  caucvgprprlem2  7643  enrer  7668  addcmpblnr  7672  mulcmpblnrlemg  7673  mulcmpblnr  7674  ltsrprg  7680  1sr  7684  m1r  7685  addclsr  7686  mulclsr  7687  addasssrg  7689  mulasssrg  7691  distrsrg  7692  m1p1sr  7693  m1m1sr  7694  lttrsr  7695  ltsosr  7697  0lt1sr  7698  0idsr  7700  1idsr  7701  00sr  7702  ltasrg  7703  recexgt0sr  7706  mulgt0sr  7711  aptisr  7712  mulextsr1lem  7713  mulextsr1  7714  archsr  7715  srpospr  7716  prsrcl  7717  prsradd  7719  prsrlt  7720  caucvgsrlemcau  7726  caucvgsrlemgt1  7728  mappsrprg  7737  map2psrprg  7738  pitonnlem1p1  7779  pitonnlem2  7780  pitonn  7781  pitoregt0  7782  pitore  7783  recnnre  7784  recidpirqlemcalc  7790  recidpirq  7791
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