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Theorem dmmulsr 10020
 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 9993 . . . 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 5432 . . 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 6859 . . 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 3741 . 2 dom ·R ⊆ (R × R)
5 0nsr 10013 . . 3 ¬ ∅ ∈ R
6 mulclsr 10018 . . 3 ((𝑥R𝑦R) → (𝑥 ·R 𝑦) ∈ R)
75, 6oprssdm 6932 . 2 (R × R) ⊆ dom ·R
84, 7eqssi 3725 1 dom ·R = (R × R)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1596  ∃wex 1817   ∈ wcel 2103  ⟨cop 4291   × cxp 5216  dom cdm 5218  (class class class)co 6765  {coprab 6766  [cec 7860   +P cpp 9796   ·P cmp 9797   ~R cer 9799  Rcnr 9800   ·R cmr 9805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-omul 7685  df-er 7862  df-ec 7864  df-qs 7868  df-ni 9807  df-pli 9808  df-mi 9809  df-lti 9810  df-plpq 9843  df-mpq 9844  df-ltpq 9845  df-enq 9846  df-nq 9847  df-erq 9848  df-plq 9849  df-mq 9850  df-1nq 9851  df-rq 9852  df-ltnq 9853  df-np 9916  df-plp 9918  df-mp 9919  df-ltp 9920  df-enr 9990  df-nr 9991  df-mr 9993 This theorem is referenced by:  mulcomsr  10023  mulasssr  10024  distrsr  10025
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