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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 9917 | . . . 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 5357 | . . 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 6784 | . . 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 3668 | . 2 ⊢ dom +R ⊆ (R × R) |
5 | 0nsr 9938 | . . 3 ⊢ ¬ ∅ ∈ R | |
6 | addclsr 9942 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 𝑦) ∈ R) | |
7 | 5, 6 | oprssdm 6857 | . 2 ⊢ (R × R) ⊆ dom +R |
8 | 4, 7 | eqssi 3652 | 1 ⊢ dom +R = (R × R) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1523 ∃wex 1744 ∈ wcel 2030 〈cop 4216 × cxp 5141 dom cdm 5143 (class class class)co 6690 {coprab 6691 [cec 7785 +P cpp 9721 ~R cer 9724 Rcnr 9725 +R cplr 9729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-omul 7610 df-er 7787 df-ec 7789 df-qs 7793 df-ni 9732 df-pli 9733 df-mi 9734 df-lti 9735 df-plpq 9768 df-mpq 9769 df-ltpq 9770 df-enq 9771 df-nq 9772 df-erq 9773 df-plq 9774 df-mq 9775 df-1nq 9776 df-rq 9777 df-ltnq 9778 df-np 9841 df-plp 9843 df-ltp 9845 df-enr 9915 df-nr 9916 df-plr 9917 |
This theorem is referenced by: addcomsr 9946 addasssr 9947 distrsr 9950 ltasr 9959 |
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