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Theorem mulclpr 9827
Description: Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
Assertion
Ref Expression
mulclpr ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)

Proof of Theorem mulclpr
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mp 9791 . 2 ·P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 ·Q 𝑧)})
2 mulclnq 9754 . 2 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
3 ltmnq 9779 . 2 (Q → (𝑓 <Q 𝑔 ↔ ( ·Q 𝑓) <Q ( ·Q 𝑔)))
4 mulcomnq 9760 . 2 (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥)
5 mulclprlem 9826 . 2 ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 ·Q ) → 𝑥 ∈ (𝐴 ·P 𝐵)))
61, 2, 3, 4, 5genpcl 9815 1 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1988  (class class class)co 6635   ·Q cmq 9663  Pcnp 9666   ·P cmp 9669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-omul 7550  df-er 7727  df-ni 9679  df-mi 9681  df-lti 9682  df-mpq 9716  df-ltpq 9717  df-enq 9718  df-nq 9719  df-erq 9720  df-mq 9722  df-1nq 9723  df-rq 9724  df-ltnq 9725  df-np 9788  df-mp 9791
This theorem is referenced by:  mulasspr  9831  distrlem1pr  9832  distrlem4pr  9833  distrlem5pr  9834  mulcmpblnr  9877  mulclsr  9890  mulasssr  9896  distrsr  9897  m1m1sr  9899  1idsr  9904  00sr  9905  recexsrlem  9909  mulgt0sr  9911
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