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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 11048 | . 2 ⊢ R = ((P × P) / ~R ) | |
2 | oveq1 7409 | . . 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 2740 | . 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R ↔ (𝐴 +R 0R) = 𝐴)) |
5 | df-0r 11052 | . . . 4 ⊢ 0R = [⟨1P, 1P⟩] ~R | |
6 | 5 | oveq2i 7413 | . . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R +R 0R) = ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) |
7 | 1pr 11007 | . . . . 5 ⊢ 1P ∈ P | |
8 | addsrpr 11067 | . . . . 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 702 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R ) |
10 | addclpr 11010 | . . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P) | |
11 | 7, 10 | mpan2 688 | . . . . . 6 ⊢ (𝑥 ∈ P → (𝑥 +P 1P) ∈ P) |
12 | addclpr 11010 | . . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 +P 1P) ∈ P) | |
13 | 7, 12 | mpan2 688 | . . . . . 6 ⊢ (𝑦 ∈ P → (𝑦 +P 1P) ∈ P) |
14 | 11, 13 | anim12i 612 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) |
15 | vex 3470 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
16 | vex 3470 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
17 | 7 | elexi 3486 | . . . . . . 7 ⊢ 1P ∈ V |
18 | addcompr 11013 | . . . . . . 7 ⊢ (𝑧 +P 𝑤) = (𝑤 +P 𝑧) | |
19 | addasspr 11014 | . . . . . . 7 ⊢ ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)) | |
20 | 15, 16, 17, 18, 19 | caov12 7629 | . . . . . 6 ⊢ (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P)) |
21 | enreceq 11058 | . . . . . 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 684 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → [⟨𝑥, 𝑦⟩] ~R = [⟨(𝑥 +P 1P), (𝑦 +P 1P)⟩] ~R ) |
24 | 9, 23 | eqtr4d 2767 | . . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) = [⟨𝑥, 𝑦⟩] ~R ) |
25 | 6, 24 | eqtrid 2776 | . 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([⟨𝑥, 𝑦⟩] ~R +R 0R) = [⟨𝑥, 𝑦⟩] ~R ) |
26 | 1, 4, 25 | ecoptocl 8798 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⟨cop 4627 (class class class)co 7402 [cec 8698 Pcnp 10851 1Pc1p 10852 +P cpp 10853 ~R cer 10856 Rcnr 10857 0Rc0r 10858 +R cplr 10861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-omul 8467 df-er 8700 df-ec 8702 df-qs 8706 df-ni 10864 df-pli 10865 df-mi 10866 df-lti 10867 df-plpq 10900 df-mpq 10901 df-ltpq 10902 df-enq 10903 df-nq 10904 df-erq 10905 df-plq 10906 df-mq 10907 df-1nq 10908 df-rq 10909 df-ltnq 10910 df-np 10973 df-1p 10974 df-plp 10975 df-ltp 10977 df-enr 11047 df-nr 11048 df-plr 11049 df-0r 11052 |
This theorem is referenced by: addgt0sr 11096 sqgt0sr 11098 map2psrpr 11102 supsrlem 11103 addresr 11130 mulresr 11131 axi2m1 11151 axcnre 11156 |
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