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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 10467 | . . 3 ⊢ R = ((P × P) / ~R ) | |
2 | oveq1 7142 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → ([〈𝑥, 𝑦〉] ~R +R [〈𝑧, 𝑤〉] ~R ) = (𝐴 +R [〈𝑧, 𝑤〉] ~R )) | |
3 | 2 | eleq1d 2874 | . . 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 7143 | . . . 4 ⊢ ([〈𝑧, 𝑤〉] ~R = 𝐵 → (𝐴 +R [〈𝑧, 𝑤〉] ~R ) = (𝐴 +R 𝐵)) | |
5 | 4 | eleq1d 2874 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ~R = 𝐵 → ((𝐴 +R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 +R 𝐵) ∈ ((P × P) / ~R ))) |
6 | addsrpr 10486 | . . . 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 10429 | . . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 +P 𝑧) ∈ P) | |
8 | addclpr 10429 | . . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 +P 𝑤) ∈ P) | |
9 | 7, 8 | anim12i 615 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P)) |
10 | 9 | an4s 659 | . . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P)) |
11 | opelxpi 5556 | . . . . 5 ⊢ (((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P) → 〈(𝑥 +P 𝑧), (𝑦 +P 𝑤)〉 ∈ (P × P)) | |
12 | enrex 10478 | . . . . . 6 ⊢ ~R ∈ V | |
13 | 12 | ecelqsi 8336 | . . . . 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 2890 | . . 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 8371 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ ((P × P) / ~R )) |
17 | 16, 1 | eleqtrrdi 2901 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ R) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 〈cop 4531 × cxp 5517 (class class class)co 7135 [cec 8270 / cqs 8271 Pcnp 10270 +P cpp 10272 ~R cer 10275 Rcnr 10276 +R cplr 10280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 df-er 8272 df-ec 8274 df-qs 8278 df-ni 10283 df-pli 10284 df-mi 10285 df-lti 10286 df-plpq 10319 df-mpq 10320 df-ltpq 10321 df-enq 10322 df-nq 10323 df-erq 10324 df-plq 10325 df-mq 10326 df-1nq 10327 df-rq 10328 df-ltnq 10329 df-np 10392 df-plp 10394 df-ltp 10396 df-enr 10466 df-nr 10467 df-plr 10468 |
This theorem is referenced by: dmaddsr 10496 map2psrpr 10521 axaddf 10556 axmulf 10557 axaddrcl 10563 axaddass 10567 axmulass 10568 axdistr 10569 |
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