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Mirrors > Home > ILE Home > Th. List > 0idsr | GIF version |
Description: The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) |
Ref | Expression |
---|---|
0idsr | ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 7723 | . 2 ⊢ R = ((P × P) / ~R ) | |
2 | oveq1 5879 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → ([〈𝑥, 𝑦〉] ~R +R 0R) = (𝐴 +R 0R)) | |
3 | id 19 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → [〈𝑥, 𝑦〉] ~R = 𝐴) | |
4 | 2, 3 | eqeq12d 2192 | . 2 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → (([〈𝑥, 𝑦〉] ~R +R 0R) = [〈𝑥, 𝑦〉] ~R ↔ (𝐴 +R 0R) = 𝐴)) |
5 | df-0r 7727 | . . . 4 ⊢ 0R = [〈1P, 1P〉] ~R | |
6 | 5 | oveq2i 5883 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R +R 0R) = ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) |
7 | 1pr 7550 | . . . . 5 ⊢ 1P ∈ P | |
8 | addsrpr 7741 | . . . . 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 439 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ) |
10 | simpl 109 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 𝑥 ∈ P) | |
11 | simpr 110 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 𝑦 ∈ P) | |
12 | 7 | a1i 9 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 1P ∈ P) |
13 | addcomprg 7574 | . . . . . . 7 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧)) | |
14 | 13 | adantl 277 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧)) |
15 | addassprg 7575 | . . . . . . 7 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P ∧ 𝑣 ∈ P) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣))) | |
16 | 15 | adantl 277 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P ∧ 𝑣 ∈ P)) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣))) |
17 | 10, 11, 12, 14, 16 | caov12d 6053 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P))) |
18 | addclpr 7533 | . . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P) | |
19 | 7, 18 | mpan2 425 | . . . . . . 7 ⊢ (𝑥 ∈ P → (𝑥 +P 1P) ∈ P) |
20 | addclpr 7533 | . . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 +P 1P) ∈ P) | |
21 | 7, 20 | mpan2 425 | . . . . . . 7 ⊢ (𝑦 ∈ P → (𝑦 +P 1P) ∈ P) |
22 | 19, 21 | anim12i 338 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) |
23 | enreceq 7732 | . . . . . 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)))) | |
24 | 22, 23 | mpdan 421 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ↔ (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P)))) |
25 | 17, 24 | mpbird 167 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → [〈𝑥, 𝑦〉] ~R = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ) |
26 | 9, 25 | eqtr4d 2213 | . . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) = [〈𝑥, 𝑦〉] ~R ) |
27 | 6, 26 | eqtrid 2222 | . 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R 0R) = [〈𝑥, 𝑦〉] ~R ) |
28 | 1, 4, 27 | ecoptocl 6619 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 〈cop 3595 (class class class)co 5872 [cec 6530 Pcnp 7287 1Pc1p 7288 +P cpp 7289 ~R cer 7292 Rcnr 7293 0Rc0r 7294 +R cplr 7297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-eprel 4288 df-id 4292 df-po 4295 df-iso 4296 df-iord 4365 df-on 4367 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-irdg 6368 df-1o 6414 df-2o 6415 df-oadd 6418 df-omul 6419 df-er 6532 df-ec 6534 df-qs 6538 df-ni 7300 df-pli 7301 df-mi 7302 df-lti 7303 df-plpq 7340 df-mpq 7341 df-enq 7343 df-nqqs 7344 df-plqqs 7345 df-mqqs 7346 df-1nqqs 7347 df-rq 7348 df-ltnqqs 7349 df-enq0 7420 df-nq0 7421 df-0nq0 7422 df-plq0 7423 df-mq0 7424 df-inp 7462 df-i1p 7463 df-iplp 7464 df-enr 7722 df-nr 7723 df-plr 7724 df-0r 7727 |
This theorem is referenced by: addgt0sr 7771 ltadd1sr 7772 ltm1sr 7773 caucvgsrlemoffval 7792 caucvgsrlemoffres 7796 caucvgsr 7798 map2psrprg 7801 suplocsrlempr 7803 addresr 7833 mulresr 7834 axi2m1 7871 ax0id 7874 axcnre 7877 |
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