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Mirrors > Home > MPE Home > Th. List > dmaddsr | Structured version Visualization version GIF version |
Description: Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dmaddsr | ⊢ dom +R = (R × R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-plr 10695 | . . . 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), (𝑣 +P 𝑓)〉] ~R ))} | |
2 | 1 | dmeqi 5787 | . . 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), (𝑣 +P 𝑓)〉] ~R ))} |
3 | dmoprabss 7331 | . . 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), (𝑣 +P 𝑓)〉] ~R ))} ⊆ (R × R) | |
4 | 2, 3 | eqsstri 3949 | . 2 ⊢ dom +R ⊆ (R × R) |
5 | 0nsr 10717 | . . 3 ⊢ ¬ ∅ ∈ R | |
6 | addclsr 10721 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 𝑦) ∈ R) | |
7 | 5, 6 | oprssdm 7407 | . 2 ⊢ (R × R) ⊆ dom +R |
8 | 4, 7 | eqssi 3931 | 1 ⊢ dom +R = (R × R) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2111 〈cop 4561 × cxp 5563 dom cdm 5565 (class class class)co 7231 {coprab 7232 [cec 8409 +P cpp 10499 ~R cer 10502 Rcnr 10503 +R cplr 10507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-inf2 9280 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-oadd 8226 df-omul 8227 df-er 8411 df-ec 8413 df-qs 8417 df-ni 10510 df-pli 10511 df-mi 10512 df-lti 10513 df-plpq 10546 df-mpq 10547 df-ltpq 10548 df-enq 10549 df-nq 10550 df-erq 10551 df-plq 10552 df-mq 10553 df-1nq 10554 df-rq 10555 df-ltnq 10556 df-np 10619 df-plp 10621 df-ltp 10623 df-enr 10693 df-nr 10694 df-plr 10695 |
This theorem is referenced by: addcomsr 10725 addasssr 10726 distrsr 10729 ltasr 10738 |
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