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Theorem mulasspr 9702
Description: Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
Assertion
Ref Expression
mulasspr ((𝐴 ·P 𝐵) ·P 𝐶) = (𝐴 ·P (𝐵 ·P 𝐶))

Proof of Theorem mulasspr
Dummy variables 𝑓 𝑔 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mp 9662 . 2 ·P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 ·Q 𝑧)})
2 mulclnq 9625 . 2 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) ∈ Q)
3 dmmp 9691 . 2 dom ·P = (P × P)
4 mulclpr 9698 . 2 ((𝑓P𝑔P) → (𝑓 ·P 𝑔) ∈ P)
5 mulassnq 9637 . 2 ((𝑓 ·Q 𝑔) ·Q ) = (𝑓 ·Q (𝑔 ·Q ))
61, 2, 3, 4, 5genpass 9687 1 ((𝐴 ·P 𝐵) ·P 𝐶) = (𝐴 ·P (𝐵 ·P 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  (class class class)co 6527   ·Q cmq 9534   ·P cmp 9540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-omul 7429  df-er 7606  df-ni 9550  df-mi 9552  df-lti 9553  df-mpq 9587  df-ltpq 9588  df-enq 9589  df-nq 9590  df-erq 9591  df-mq 9593  df-1nq 9594  df-rq 9595  df-ltnq 9596  df-np 9659  df-mp 9662
This theorem is referenced by:  mulasssr  9767
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