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Mirrors > Home > MPE Home > Th. List > dmmulsr | Structured version Visualization version GIF version |
Description: Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dmmulsr | ⊢ dom ·R = (R × R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mr 10168 | . . . 4 ⊢ ·R = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~R ∧ 𝑦 = [〈𝑢, 𝑓〉] ~R ) ∧ 𝑧 = [〈((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))〉] ~R ))} | |
2 | 1 | dmeqi 5528 | . . 3 ⊢ dom ·R = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~R ∧ 𝑦 = [〈𝑢, 𝑓〉] ~R ) ∧ 𝑧 = [〈((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))〉] ~R ))} |
3 | dmoprabss 6976 | . . 3 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~R ∧ 𝑦 = [〈𝑢, 𝑓〉] ~R ) ∧ 𝑧 = [〈((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))〉] ~R ))} ⊆ (R × R) | |
4 | 2, 3 | eqsstri 3831 | . 2 ⊢ dom ·R ⊆ (R × R) |
5 | 0nsr 10188 | . . 3 ⊢ ¬ ∅ ∈ R | |
6 | mulclsr 10193 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 ·R 𝑦) ∈ R) | |
7 | 5, 6 | oprssdm 7049 | . 2 ⊢ (R × R) ⊆ dom ·R |
8 | 4, 7 | eqssi 3814 | 1 ⊢ dom ·R = (R × R) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 385 = wceq 1653 ∃wex 1875 ∈ wcel 2157 〈cop 4374 × cxp 5310 dom cdm 5312 (class class class)co 6878 {coprab 6879 [cec 7980 +P cpp 9971 ·P cmp 9972 ~R cer 9974 Rcnr 9975 ·R cmr 9980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-omul 7804 df-er 7982 df-ec 7984 df-qs 7988 df-ni 9982 df-pli 9983 df-mi 9984 df-lti 9985 df-plpq 10018 df-mpq 10019 df-ltpq 10020 df-enq 10021 df-nq 10022 df-erq 10023 df-plq 10024 df-mq 10025 df-1nq 10026 df-rq 10027 df-ltnq 10028 df-np 10091 df-plp 10093 df-mp 10094 df-ltp 10095 df-enr 10165 df-nr 10166 df-mr 10168 |
This theorem is referenced by: mulcomsr 10198 mulasssr 10199 distrsr 10200 |
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