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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 10842 | . . . 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 5817 | . . 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 7397 | . . 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 3957 | . 2 ⊢ dom ·R ⊆ (R × R) |
5 | 0nsr 10863 | . . 3 ⊢ ¬ ∅ ∈ R | |
6 | mulclsr 10868 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 ·R 𝑦) ∈ R) | |
7 | 5, 6 | oprssdm 7473 | . 2 ⊢ (R × R) ⊆ dom ·R |
8 | 4, 7 | eqssi 3939 | 1 ⊢ dom ·R = (R × R) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∃wex 1777 ∈ wcel 2101 〈cop 4570 × cxp 5589 dom cdm 5591 (class class class)co 7295 {coprab 7296 [cec 8516 +P cpp 10645 ·P cmp 10646 ~R cer 10648 Rcnr 10649 ·R cmr 10654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-inf2 9427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-oadd 8321 df-omul 8322 df-er 8518 df-ec 8520 df-qs 8524 df-ni 10656 df-pli 10657 df-mi 10658 df-lti 10659 df-plpq 10692 df-mpq 10693 df-ltpq 10694 df-enq 10695 df-nq 10696 df-erq 10697 df-plq 10698 df-mq 10699 df-1nq 10700 df-rq 10701 df-ltnq 10702 df-np 10765 df-plp 10767 df-mp 10768 df-ltp 10769 df-enr 10839 df-nr 10840 df-mr 10842 |
This theorem is referenced by: mulcomsr 10873 mulasssr 10874 distrsr 10875 |
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