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| Mirrors > Home > MPE Home > Th. List > 0idsr | Structured version Visualization version GIF version | ||
| Description: The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| 0idsr | ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-nr 11097 | . 2 ⊢ R = ((P × P) / ~R ) | |
| 2 | oveq1 7439 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → ([〈𝑥, 𝑦〉] ~R +R 0R) = (𝐴 +R 0R)) | |
| 3 | id 22 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → [〈𝑥, 𝑦〉] ~R = 𝐴) | |
| 4 | 2, 3 | eqeq12d 2752 | . 2 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → (([〈𝑥, 𝑦〉] ~R +R 0R) = [〈𝑥, 𝑦〉] ~R ↔ (𝐴 +R 0R) = 𝐴)) | 
| 5 | df-0r 11101 | . . . 4 ⊢ 0R = [〈1P, 1P〉] ~R | |
| 6 | 5 | oveq2i 7443 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R +R 0R) = ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) | 
| 7 | 1pr 11056 | . . . . 5 ⊢ 1P ∈ P | |
| 8 | addsrpr 11116 | . . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (1P ∈ P ∧ 1P ∈ P)) → ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ) | |
| 9 | 7, 7, 8 | mpanr12 705 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ) | 
| 10 | addclpr 11059 | . . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P) | |
| 11 | 7, 10 | mpan2 691 | . . . . . 6 ⊢ (𝑥 ∈ P → (𝑥 +P 1P) ∈ P) | 
| 12 | addclpr 11059 | . . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 +P 1P) ∈ P) | |
| 13 | 7, 12 | mpan2 691 | . . . . . 6 ⊢ (𝑦 ∈ P → (𝑦 +P 1P) ∈ P) | 
| 14 | 11, 13 | anim12i 613 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) | 
| 15 | vex 3483 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 16 | vex 3483 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 17 | 7 | elexi 3502 | . . . . . . 7 ⊢ 1P ∈ V | 
| 18 | addcompr 11062 | . . . . . . 7 ⊢ (𝑧 +P 𝑤) = (𝑤 +P 𝑧) | |
| 19 | addasspr 11063 | . . . . . . 7 ⊢ ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)) | |
| 20 | 15, 16, 17, 18, 19 | caov12 7662 | . . . . . 6 ⊢ (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P)) | 
| 21 | enreceq 11107 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) → ([〈𝑥, 𝑦〉] ~R = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ↔ (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P)))) | |
| 22 | 20, 21 | mpbiri 258 | . . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) → [〈𝑥, 𝑦〉] ~R = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ) | 
| 23 | 14, 22 | mpdan 687 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → [〈𝑥, 𝑦〉] ~R = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ) | 
| 24 | 9, 23 | eqtr4d 2779 | . . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) = [〈𝑥, 𝑦〉] ~R ) | 
| 25 | 6, 24 | eqtrid 2788 | . 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R 0R) = [〈𝑥, 𝑦〉] ~R ) | 
| 26 | 1, 4, 25 | ecoptocl 8848 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 〈cop 4631 (class class class)co 7432 [cec 8744 Pcnp 10900 1Pc1p 10901 +P cpp 10902 ~R cer 10905 Rcnr 10906 0Rc0r 10907 +R cplr 10910 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-omul 8512 df-er 8746 df-ec 8748 df-qs 8752 df-ni 10913 df-pli 10914 df-mi 10915 df-lti 10916 df-plpq 10949 df-mpq 10950 df-ltpq 10951 df-enq 10952 df-nq 10953 df-erq 10954 df-plq 10955 df-mq 10956 df-1nq 10957 df-rq 10958 df-ltnq 10959 df-np 11022 df-1p 11023 df-plp 11024 df-ltp 11026 df-enr 11096 df-nr 11097 df-plr 11098 df-0r 11101 | 
| This theorem is referenced by: addgt0sr 11145 sqgt0sr 11147 map2psrpr 11151 supsrlem 11152 addresr 11179 mulresr 11180 axi2m1 11200 axcnre 11205 | 
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