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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 11094 | . 2 ⊢ R = ((P × P) / ~R ) | |
2 | oveq1 7438 | . . 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 2751 | . 2 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → (([〈𝑥, 𝑦〉] ~R +R 0R) = [〈𝑥, 𝑦〉] ~R ↔ (𝐴 +R 0R) = 𝐴)) |
5 | df-0r 11098 | . . . 4 ⊢ 0R = [〈1P, 1P〉] ~R | |
6 | 5 | oveq2i 7442 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R +R 0R) = ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) |
7 | 1pr 11053 | . . . . 5 ⊢ 1P ∈ P | |
8 | addsrpr 11113 | . . . . 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 11056 | . . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P) | |
11 | 7, 10 | mpan2 691 | . . . . . 6 ⊢ (𝑥 ∈ P → (𝑥 +P 1P) ∈ P) |
12 | addclpr 11056 | . . . . . . 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 3482 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
16 | vex 3482 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
17 | 7 | elexi 3501 | . . . . . . 7 ⊢ 1P ∈ V |
18 | addcompr 11059 | . . . . . . 7 ⊢ (𝑧 +P 𝑤) = (𝑤 +P 𝑧) | |
19 | addasspr 11060 | . . . . . . 7 ⊢ ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)) | |
20 | 15, 16, 17, 18, 19 | caov12 7661 | . . . . . 6 ⊢ (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P)) |
21 | enreceq 11104 | . . . . . 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 2778 | . . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) = [〈𝑥, 𝑦〉] ~R ) |
25 | 6, 24 | eqtrid 2787 | . 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R 0R) = [〈𝑥, 𝑦〉] ~R ) |
26 | 1, 4, 25 | ecoptocl 8846 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 〈cop 4637 (class class class)co 7431 [cec 8742 Pcnp 10897 1Pc1p 10898 +P cpp 10899 ~R cer 10902 Rcnr 10903 0Rc0r 10904 +R cplr 10907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-omul 8510 df-er 8744 df-ec 8746 df-qs 8750 df-ni 10910 df-pli 10911 df-mi 10912 df-lti 10913 df-plpq 10946 df-mpq 10947 df-ltpq 10948 df-enq 10949 df-nq 10950 df-erq 10951 df-plq 10952 df-mq 10953 df-1nq 10954 df-rq 10955 df-ltnq 10956 df-np 11019 df-1p 11020 df-plp 11021 df-ltp 11023 df-enr 11093 df-nr 11094 df-plr 11095 df-0r 11098 |
This theorem is referenced by: addgt0sr 11142 sqgt0sr 11144 map2psrpr 11148 supsrlem 11149 addresr 11176 mulresr 11177 axi2m1 11197 axcnre 11202 |
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