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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 11096 | . . 3 ⊢ R = ((P × P) / ~R ) | |
| 2 | oveq1 7438 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → ([〈𝑥, 𝑦〉] ~R +R [〈𝑧, 𝑤〉] ~R ) = (𝐴 +R [〈𝑧, 𝑤〉] ~R )) | |
| 3 | 2 | eleq1d 2826 | . . 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 7439 | . . . 4 ⊢ ([〈𝑧, 𝑤〉] ~R = 𝐵 → (𝐴 +R [〈𝑧, 𝑤〉] ~R ) = (𝐴 +R 𝐵)) | |
| 5 | 4 | eleq1d 2826 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ~R = 𝐵 → ((𝐴 +R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 +R 𝐵) ∈ ((P × P) / ~R ))) | 
| 6 | addsrpr 11115 | . . . 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 11058 | . . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 +P 𝑧) ∈ P) | |
| 8 | addclpr 11058 | . . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 +P 𝑤) ∈ P) | |
| 9 | 7, 8 | anim12i 613 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P)) | 
| 10 | 9 | an4s 660 | . . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P)) | 
| 11 | opelxpi 5722 | . . . . 5 ⊢ (((𝑥 +P 𝑧) ∈ P ∧ (𝑦 +P 𝑤) ∈ P) → 〈(𝑥 +P 𝑧), (𝑦 +P 𝑤)〉 ∈ (P × P)) | |
| 12 | enrex 11107 | . . . . . 6 ⊢ ~R ∈ V | |
| 13 | 12 | ecelqsi 8813 | . . . . 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 2841 | . . 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 8848 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ ((P × P) / ~R )) | 
| 17 | 16, 1 | eleqtrrdi 2852 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ R) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 × cxp 5683 (class class class)co 7431 [cec 8743 / cqs 8744 Pcnp 10899 +P cpp 10901 ~R cer 10904 Rcnr 10905 +R cplr 10909 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-ec 8747 df-qs 8751 df-ni 10912 df-pli 10913 df-mi 10914 df-lti 10915 df-plpq 10948 df-mpq 10949 df-ltpq 10950 df-enq 10951 df-nq 10952 df-erq 10953 df-plq 10954 df-mq 10955 df-1nq 10956 df-rq 10957 df-ltnq 10958 df-np 11021 df-plp 11023 df-ltp 11025 df-enr 11095 df-nr 11096 df-plr 11097 | 
| This theorem is referenced by: dmaddsr 11125 map2psrpr 11150 axaddf 11185 axmulf 11186 axaddrcl 11192 axaddass 11196 axmulass 11197 axdistr 11198 | 
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