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Theorem addclpr 7313
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 7244 . . . 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 7287 . . 3 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3 addclnq 7151 . . . 4 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
41, 3genpml 7293 . . 3 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))
51, 3genpmu 7294 . . 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 7176 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧 +Q 𝑥) <Q (𝑧 +Q 𝑦)))
8 addcomnqg 7157 . . . . 5 ((𝑥Q𝑦Q) → (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥))
9 addnqprl 7305 . . . . 5 ((((𝐴P𝑔 ∈ (1st𝐴)) ∧ (𝐵P ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → 𝑥 ∈ (1st ‘(𝐴 +P 𝐵))))
101, 3, 7, 8, 9genprndl 7297 . . . 4 ((𝐴P𝐵P) → ∀𝑞Q (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴 +P 𝐵)))))
11 addnqpru 7306 . . . . 5 ((((𝐴P𝑔 ∈ (2nd𝐴)) ∧ (𝐵P ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑔 +Q ) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴 +P 𝐵))))
121, 3, 7, 8, 11genprndu 7298 . . . 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 7299 . . 3 ((𝐴P𝐵P) → ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 +P 𝐵))))
15 addlocpr 7312 . . 3 ((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
1613, 14, 153jca 1146 . 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 7252 . 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 682  w3a 947  wcel 1465  wral 2393  wrex 2394  𝒫 cpw 3480   class class class wbr 3899   × cxp 4507  cfv 5093  (class class class)co 5742  1st c1st 6004  2nd c2nd 6005  Qcnq 7056   +Q cplq 7058   <Q cltq 7061  Pcnp 7067   +P cpp 7069
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244
This theorem is referenced by:  addnqprlemfl  7335  addnqprlemfu  7336  addnqpr  7337  addassprg  7355  distrlem1prl  7358  distrlem1pru  7359  distrlem4prl  7360  distrlem4pru  7361  distrprg  7364  ltaddpr  7373  ltexpri  7389  addcanprleml  7390  addcanprlemu  7391  ltaprlem  7394  ltaprg  7395  prplnqu  7396  addextpr  7397  caucvgprlemcanl  7420  cauappcvgprlemladdru  7432  cauappcvgprlemladdrl  7433  cauappcvgprlemladd  7434  cauappcvgprlem1  7435  caucvgprlemladdrl  7454  caucvgprlem1  7455  caucvgprprlemnbj  7469  caucvgprprlemopu  7475  caucvgprprlemloc  7479  caucvgprprlemexbt  7482  caucvgprprlemexb  7483  caucvgprprlemaddq  7484  caucvgprprlem2  7486  enrer  7511  addcmpblnr  7515  mulcmpblnrlemg  7516  mulcmpblnr  7517  ltsrprg  7523  1sr  7527  m1r  7528  addclsr  7529  mulclsr  7530  addasssrg  7532  mulasssrg  7534  distrsrg  7535  m1p1sr  7536  m1m1sr  7537  lttrsr  7538  ltsosr  7540  0lt1sr  7541  0idsr  7543  1idsr  7544  00sr  7545  ltasrg  7546  recexgt0sr  7549  mulgt0sr  7554  aptisr  7555  mulextsr1lem  7556  mulextsr1  7557  archsr  7558  srpospr  7559  prsrcl  7560  prsradd  7562  prsrlt  7563  caucvgsrlemcau  7569  caucvgsrlemgt1  7571  mappsrprg  7580  map2psrprg  7581  pitonnlem1p1  7622  pitonnlem2  7623  pitonn  7624  pitoregt0  7625  pitore  7626  recnnre  7627  recidpirqlemcalc  7633  recidpirq  7634
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