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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 10082 | . . . 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 5464 | . . 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 6890 | . . 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 3785 | . 2 ⊢ dom +R ⊆ (R × R) |
5 | 0nsr 10103 | . . 3 ⊢ ¬ ∅ ∈ R | |
6 | addclsr 10107 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 +R 𝑦) ∈ R) | |
7 | 5, 6 | oprssdm 6963 | . 2 ⊢ (R × R) ⊆ dom +R |
8 | 4, 7 | eqssi 3769 | 1 ⊢ dom +R = (R × R) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 = wceq 1631 ∃wex 1852 ∈ wcel 2145 〈cop 4323 × cxp 5248 dom cdm 5250 (class class class)co 6794 {coprab 6795 [cec 7895 +P cpp 9886 ~R cer 9889 Rcnr 9890 +R cplr 9894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-inf2 8703 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-1st 7316 df-2nd 7317 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-1o 7714 df-oadd 7718 df-omul 7719 df-er 7897 df-ec 7899 df-qs 7903 df-ni 9897 df-pli 9898 df-mi 9899 df-lti 9900 df-plpq 9933 df-mpq 9934 df-ltpq 9935 df-enq 9936 df-nq 9937 df-erq 9938 df-plq 9939 df-mq 9940 df-1nq 9941 df-rq 9942 df-ltnq 9943 df-np 10006 df-plp 10008 df-ltp 10010 df-enr 10080 df-nr 10081 df-plr 10082 |
This theorem is referenced by: addcomsr 10111 addasssr 10112 distrsr 10115 ltasr 10124 |
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