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Theorem mulclpr 6613
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 6510 . . . 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 6552 . . 3 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3 mulclnq 6417 . . . 4 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
41, 3genpml 6558 . . 3 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)))
51, 3genpmu 6559 . . 3 ((𝐴P𝐵P) → ∃𝑟Q 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))
62, 4, 5jca32 293 . 2 ((𝐴P𝐵P) → ((𝐴 ·P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ ∃𝑟Q 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
7 ltmnqg 6442 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧 ·Q 𝑥) <Q (𝑧 ·Q 𝑦)))
8 mulcomnqg 6424 . . . . 5 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
9 mulnqprl 6609 . . . . 5 ((((𝐴P𝑢 ∈ (1st𝐴)) ∧ (𝐵P𝑡 ∈ (1st𝐵))) ∧ 𝑥Q) → (𝑥 <Q (𝑢 ·Q 𝑡) → 𝑥 ∈ (1st ‘(𝐴 ·P 𝐵))))
101, 3, 7, 8, 9genprndl 6562 . . . 4 ((𝐴P𝐵P) → ∀𝑞Q (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st ‘(𝐴 ·P 𝐵)))))
11 mulnqpru 6610 . . . . 5 ((((𝐴P𝑢 ∈ (2nd𝐴)) ∧ (𝐵P𝑡 ∈ (2nd𝐵))) ∧ 𝑥Q) → ((𝑢 ·Q 𝑡) <Q 𝑥𝑥 ∈ (2nd ‘(𝐴 ·P 𝐵))))
121, 3, 7, 8, 11genprndu 6563 . . . 4 ((𝐴P𝐵P) → ∀𝑟Q (𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
1310, 12jca 290 . . 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 6564 . . 3 ((𝐴P𝐵P) → ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴 ·P 𝐵))))
15 mullocpr 6612 . . 3 ((𝐴P𝐵P) → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
1613, 14, 153jca 1084 . 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 6517 . 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 394 1 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  wo 629  w3a 885  wcel 1393  wral 2303  wrex 2304  𝒫 cpw 3356   class class class wbr 3760   × cxp 4304  cfv 4863  (class class class)co 5473  1st c1st 5726  2nd c2nd 5727  Qcnq 6321   ·Q cmq 6324   <Q cltq 6326  Pcnp 6332   ·P cmp 6335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3868  ax-sep 3871  ax-nul 3879  ax-pow 3923  ax-pr 3940  ax-un 4141  ax-setind 4230  ax-iinf 4272
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3577  df-int 3612  df-iun 3655  df-br 3761  df-opab 3815  df-mpt 3816  df-tr 3851  df-eprel 4022  df-id 4026  df-po 4029  df-iso 4030  df-iord 4074  df-on 4076  df-suc 4079  df-iom 4275  df-xp 4312  df-rel 4313  df-cnv 4314  df-co 4315  df-dm 4316  df-rn 4317  df-res 4318  df-ima 4319  df-iota 4828  df-fun 4865  df-fn 4866  df-f 4867  df-f1 4868  df-fo 4869  df-f1o 4870  df-fv 4871  df-ov 5476  df-oprab 5477  df-mpt2 5478  df-1st 5728  df-2nd 5729  df-recs 5881  df-irdg 5918  df-1o 5962  df-2o 5963  df-oadd 5966  df-omul 5967  df-er 6065  df-ec 6067  df-qs 6071  df-ni 6345  df-pli 6346  df-mi 6347  df-lti 6348  df-plpq 6385  df-mpq 6386  df-enq 6388  df-nqqs 6389  df-plqqs 6390  df-mqqs 6391  df-1nqqs 6392  df-rq 6393  df-ltnqqs 6394  df-enq0 6465  df-nq0 6466  df-0nq0 6467  df-plq0 6468  df-mq0 6469  df-inp 6507  df-imp 6510
This theorem is referenced by:  mulnqprlemfl  6616  mulnqprlemfu  6617  mulnqpr  6618  mulassprg  6622  distrlem1prl  6623  distrlem1pru  6624  distrlem4prl  6625  distrlem4pru  6626  distrlem5prl  6627  distrlem5pru  6628  distrprg  6629  1idpr  6633  recexprlemex  6678  ltmprr  6683  mulcmpblnrlemg  6768  mulcmpblnr  6769  mulclsr  6782  mulcomsrg  6785  mulasssrg  6786  distrsrg  6787  m1m1sr  6789  1idsr  6796  00sr  6797  recexgt0sr  6801  mulgt0sr  6805  mulextsr1lem  6807  mulextsr1  6808  recidpirq  6877
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