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Mirrors > Home > MPE Home > Th. List > addclsr | Structured version Visualization version GIF version |
Description: Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addclsr | ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 9916 | . . 3 ⊢ R = ((P × P) / ~R ) | |
2 | oveq1 6697 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → ([〈𝑥, 𝑦〉] ~R +R [〈𝑧, 𝑤〉] ~R ) = (𝐴 +R [〈𝑧, 𝑤〉] ~R )) | |
3 | 2 | eleq1d 2715 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → (([〈𝑥, 𝑦〉] ~R +R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 +R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R ))) |
4 | oveq2 6698 | . . . 4 ⊢ ([〈𝑧, 𝑤〉] ~R = 𝐵 → (𝐴 +R [〈𝑧, 𝑤〉] ~R ) = (𝐴 +R 𝐵)) | |
5 | 4 | eleq1d 2715 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ~R = 𝐵 → ((𝐴 +R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 +R 𝐵) ∈ ((P × P) / ~R ))) |
6 | addsrpr 9934 | . . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([〈𝑥, 𝑦〉] ~R +R [〈𝑧, 𝑤〉] ~R ) = [〈(𝑥 +P 𝑧), (𝑦 +P 𝑤)〉] ~R ) | |
7 | addclpr 9878 | . . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 +P 𝑧) ∈ P) | |
8 | addclpr 9878 | . . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 +P 𝑤) ∈ P) | |
9 | 7, 8 | anim12i 589 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P)) |
10 | 9 | an4s 886 | . . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P)) |
11 | opelxpi 5182 | . . . . 5 ⊢ (((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P) → 〈(𝑥 +P 𝑧), (𝑦 +P 𝑤)〉 ∈ (P × P)) | |
12 | enrex 9926 | . . . . . 6 ⊢ ~R ∈ V | |
13 | 12 | ecelqsi 7846 | . . . . 5 ⊢ (〈(𝑥 +P 𝑧), (𝑦 +P 𝑤)〉 ∈ (P × P) → [〈(𝑥 +P 𝑧), (𝑦 +P 𝑤)〉] ~R ∈ ((P × P) / ~R )) |
14 | 10, 11, 13 | 3syl 18 | . . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → [〈(𝑥 +P 𝑧), (𝑦 +P 𝑤)〉] ~R ∈ ((P × P) / ~R )) |
15 | 6, 14 | eqeltrd 2730 | . . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([〈𝑥, 𝑦〉] ~R +R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R )) |
16 | 1, 3, 5, 15 | 2ecoptocl 7881 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ ((P × P) / ~R )) |
17 | 16, 1 | syl6eleqr 2741 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ R) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 〈cop 4216 × cxp 5141 (class class class)co 6690 [cec 7785 / cqs 7786 Pcnp 9719 +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: dmaddsr 9944 map2psrpr 9969 axaddf 10004 axmulf 10005 axaddrcl 10011 axaddass 10015 axmulass 10016 axdistr 10017 |
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