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Theorem mulclpr 6813
Description: Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)
Assertion
Ref Expression
mulclpr ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)

Proof of Theorem mulclpr
Dummy variables 𝑞 𝑟 𝑡 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-imp 6710 . . . 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 6752 . . 3 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3 mulclnq 6617 . . . 4 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
41, 3genpml 6758 . . 3 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)))
51, 3genpmu 6759 . . 3 ((𝐴P𝐵P) → ∃𝑟Q 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))
62, 4, 5jca32 303 . 2 ((𝐴P𝐵P) → ((𝐴 ·P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ ∃𝑟Q 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
7 ltmnqg 6642 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧 ·Q 𝑥) <Q (𝑧 ·Q 𝑦)))
8 mulcomnqg 6624 . . . . 5 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
9 mulnqprl 6809 . . . . 5 ((((𝐴P𝑢 ∈ (1st𝐴)) ∧ (𝐵P𝑡 ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑢 ·Q 𝑡) → 𝑥 ∈ (1st ‘(𝐴 ·P 𝐵))))
101, 3, 7, 8, 9genprndl 6762 . . . 4 ((𝐴P𝐵P) → ∀𝑞Q (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴 ·P 𝐵)))))
11 mulnqpru 6810 . . . . 5 ((((𝐴P𝑢 ∈ (2nd𝐴)) ∧ (𝐵P𝑡 ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑢 ·Q 𝑡) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴 ·P 𝐵))))
121, 3, 7, 8, 11genprndu 6763 . . . 4 ((𝐴P𝐵P) → ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
1310, 12jca 300 . . 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 6764 . . 3 ((𝐴P𝐵P) → ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 ·P 𝐵))))
15 mullocpr 6812 . . 3 ((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
1613, 14, 153jca 1119 . 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 6717 . 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 408 1 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3a 920  wcel 1434  wral 2349  wrex 2350  𝒫 cpw 3384   class class class wbr 3787   × cxp 4363  cfv 4926  (class class class)co 5537  1st c1st 5790  2nd c2nd 5791  Qcnq 6521   ·Q cmq 6524   <Q cltq 6526  Pcnp 6532   ·P cmp 6535
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-2o 6060  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594  df-enq0 6665  df-nq0 6666  df-0nq0 6667  df-plq0 6668  df-mq0 6669  df-inp 6707  df-imp 6710
This theorem is referenced by:  mulnqprlemfl  6816  mulnqprlemfu  6817  mulnqpr  6818  mulassprg  6822  distrlem1prl  6823  distrlem1pru  6824  distrlem4prl  6825  distrlem4pru  6826  distrlem5prl  6827  distrlem5pru  6828  distrprg  6829  1idpr  6833  recexprlemex  6878  ltmprr  6883  mulcmpblnrlemg  6968  mulcmpblnr  6969  mulclsr  6982  mulcomsrg  6985  mulasssrg  6986  distrsrg  6987  m1m1sr  6989  1idsr  6996  00sr  6997  recexgt0sr  7001  mulgt0sr  7005  mulextsr1lem  7007  mulextsr1  7008  recidpirq  7077
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