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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 10825 | . . . 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 5812 | . . 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 7372 | . . 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 3960 | . 2 ⊢ dom ·R ⊆ (R × R) |
5 | 0nsr 10846 | . . 3 ⊢ ¬ ∅ ∈ R | |
6 | mulclsr 10851 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 ·R 𝑦) ∈ R) | |
7 | 5, 6 | oprssdm 7448 | . 2 ⊢ (R × R) ⊆ dom ·R |
8 | 4, 7 | eqssi 3942 | 1 ⊢ dom ·R = (R × R) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1542 ∃wex 1786 ∈ wcel 2110 〈cop 4573 × cxp 5588 dom cdm 5590 (class class class)co 7272 {coprab 7273 [cec 8488 +P cpp 10628 ·P cmp 10629 ~R cer 10631 Rcnr 10632 ·R cmr 10637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-inf2 9387 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-oadd 8293 df-omul 8294 df-er 8490 df-ec 8492 df-qs 8496 df-ni 10639 df-pli 10640 df-mi 10641 df-lti 10642 df-plpq 10675 df-mpq 10676 df-ltpq 10677 df-enq 10678 df-nq 10679 df-erq 10680 df-plq 10681 df-mq 10682 df-1nq 10683 df-rq 10684 df-ltnq 10685 df-np 10748 df-plp 10750 df-mp 10751 df-ltp 10752 df-enr 10822 df-nr 10823 df-mr 10825 |
This theorem is referenced by: mulcomsr 10856 mulasssr 10857 distrsr 10858 |
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