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Theorem dmmulsr 9760
Description: Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
dmmulsr dom ·R = (R × R)

Proof of Theorem dmmulsr
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mr 9733 . . . 4 ·R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R ))}
21dmeqi 5231 . . 3 dom ·R = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R ))}
3 dmoprabss 6615 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R ))} ⊆ (R × R)
42, 3eqsstri 3594 . 2 dom ·R ⊆ (R × R)
5 0nsr 9753 . . 3 ¬ ∅ ∈ R
6 mulclsr 9758 . . 3 ((𝑥R𝑦R) → (𝑥 ·R 𝑦) ∈ R)
75, 6oprssdm 6687 . 2 (R × R) ⊆ dom ·R
84, 7eqssi 3580 1 dom ·R = (R × R)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wex 1694  wcel 1976  cop 4127   × cxp 5023  dom cdm 5025  (class class class)co 6524  {coprab 6525  [cec 7601   +P cpp 9536   ·P cmp 9537   ~R cer 9539  Rcnr 9540   ·R cmr 9545
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 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395
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 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-omul 7426  df-er 7603  df-ec 7605  df-qs 7609  df-ni 9547  df-pli 9548  df-mi 9549  df-lti 9550  df-plpq 9583  df-mpq 9584  df-ltpq 9585  df-enq 9586  df-nq 9587  df-erq 9588  df-plq 9589  df-mq 9590  df-1nq 9591  df-rq 9592  df-ltnq 9593  df-np 9656  df-plp 9658  df-mp 9659  df-ltp 9660  df-enr 9730  df-nr 9731  df-mr 9733
This theorem is referenced by:  mulcomsr  9763  mulasssr  9764  distrsr  9765
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