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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 10503 | . . . 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 5737 | . . 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 7243 | . . 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 3922 | . 2 ⊢ dom ·R ⊆ (R × R) |
5 | 0nsr 10524 | . . 3 ⊢ ¬ ∅ ∈ R | |
6 | mulclsr 10529 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 ·R 𝑦) ∈ R) | |
7 | 5, 6 | oprssdm 7318 | . 2 ⊢ (R × R) ⊆ dom ·R |
8 | 4, 7 | eqssi 3904 | 1 ⊢ dom ·R = (R × R) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 400 = wceq 1539 ∃wex 1782 ∈ wcel 2112 〈cop 4521 × cxp 5515 dom cdm 5517 (class class class)co 7143 {coprab 7144 [cec 8290 +P cpp 10306 ·P cmp 10307 ~R cer 10309 Rcnr 10310 ·R cmr 10315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-inf2 9122 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-omul 8110 df-er 8292 df-ec 8294 df-qs 8298 df-ni 10317 df-pli 10318 df-mi 10319 df-lti 10320 df-plpq 10353 df-mpq 10354 df-ltpq 10355 df-enq 10356 df-nq 10357 df-erq 10358 df-plq 10359 df-mq 10360 df-1nq 10361 df-rq 10362 df-ltnq 10363 df-np 10426 df-plp 10428 df-mp 10429 df-ltp 10430 df-enr 10500 df-nr 10501 df-mr 10503 |
This theorem is referenced by: mulcomsr 10534 mulasssr 10535 distrsr 10536 |
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