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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 10913 | . 2 ⊢ R = ((P × P) / ~R ) | |
2 | oveq1 7344 | . . 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 10917 | . . . 4 ⊢ 0R = [⟨1P, 1P⟩] ~R | |
6 | 5 | oveq2i 7348 | . . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R +R 0R) = ([⟨𝑥, 𝑦⟩] ~R +R [⟨1P, 1P⟩] ~R ) |
7 | 1pr 10872 | . . . . 5 ⊢ 1P ∈ P | |
8 | addsrpr 10932 | . . . . 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 10875 | . . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P) | |
11 | 7, 10 | mpan2 688 | . . . . . 6 ⊢ (𝑥 ∈ P → (𝑥 +P 1P) ∈ P) |
12 | addclpr 10875 | . . . . . . 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 613 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) |
15 | vex 3445 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
16 | vex 3445 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
17 | 7 | elexi 3460 | . . . . . . 7 ⊢ 1P ∈ V |
18 | addcompr 10878 | . . . . . . 7 ⊢ (𝑧 +P 𝑤) = (𝑤 +P 𝑧) | |
19 | addasspr 10879 | . . . . . . 7 ⊢ ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)) | |
20 | 15, 16, 17, 18, 19 | caov12 7562 | . . . . . 6 ⊢ (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P)) |
21 | enreceq 10923 | . . . . . 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 257 | . . . . 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 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 8667 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ⟨cop 4579 (class class class)co 7337 [cec 8567 Pcnp 10716 1Pc1p 10717 +P cpp 10718 ~R cer 10721 Rcnr 10722 0Rc0r 10723 +R cplr 10726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-inf2 9498 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-oadd 8371 df-omul 8372 df-er 8569 df-ec 8571 df-qs 8575 df-ni 10729 df-pli 10730 df-mi 10731 df-lti 10732 df-plpq 10765 df-mpq 10766 df-ltpq 10767 df-enq 10768 df-nq 10769 df-erq 10770 df-plq 10771 df-mq 10772 df-1nq 10773 df-rq 10774 df-ltnq 10775 df-np 10838 df-1p 10839 df-plp 10840 df-ltp 10842 df-enr 10912 df-nr 10913 df-plr 10914 df-0r 10917 |
This theorem is referenced by: addgt0sr 10961 sqgt0sr 10963 map2psrpr 10967 supsrlem 10968 addresr 10995 mulresr 10996 axi2m1 11016 axcnre 11021 |
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