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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 6966 | . 2 ⊢ R = ((P × P) / ~R ) | |
2 | oveq1 5550 | . . 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 2096 | . 2 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → (([〈𝑥, 𝑦〉] ~R +R 0R) = [〈𝑥, 𝑦〉] ~R ↔ (𝐴 +R 0R) = 𝐴)) |
5 | df-0r 6970 | . . . 4 ⊢ 0R = [〈1P, 1P〉] ~R | |
6 | 5 | oveq2i 5554 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R +R 0R) = ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) |
7 | 1pr 6806 | . . . . 5 ⊢ 1P ∈ P | |
8 | addsrpr 6984 | . . . . 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 430 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ) |
10 | simpl 107 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 𝑥 ∈ P) | |
11 | simpr 108 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 𝑦 ∈ P) | |
12 | 7 | a1i 9 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → 1P ∈ P) |
13 | addcomprg 6830 | . . . . . . 7 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧)) | |
14 | 13 | adantl 271 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑧 +P 𝑤) = (𝑤 +P 𝑧)) |
15 | addassprg 6831 | . . . . . . 7 ⊢ ((𝑧 ∈ P ∧ 𝑤 ∈ P ∧ 𝑣 ∈ P) → ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣))) | |
16 | 15 | adantl 271 | . . . . . 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 5713 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P))) |
18 | addclpr 6789 | . . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P) | |
19 | 7, 18 | mpan2 416 | . . . . . . 7 ⊢ (𝑥 ∈ P → (𝑥 +P 1P) ∈ P) |
20 | addclpr 6789 | . . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 +P 1P) ∈ P) | |
21 | 7, 20 | mpan2 416 | . . . . . . 7 ⊢ (𝑦 ∈ P → (𝑦 +P 1P) ∈ P) |
22 | 19, 21 | anim12i 331 | . . . . . 6 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) |
23 | enreceq 6975 | . . . . . 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 412 | . . . . 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 165 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → [〈𝑥, 𝑦〉] ~R = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ) |
26 | 9, 25 | eqtr4d 2117 | . . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) = [〈𝑥, 𝑦〉] ~R ) |
27 | 6, 26 | syl5eq 2126 | . 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R 0R) = [〈𝑥, 𝑦〉] ~R ) |
28 | 1, 4, 27 | ecoptocl 6259 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 〈cop 3409 (class class class)co 5543 [cec 6170 Pcnp 6543 1Pc1p 6544 +P cpp 6545 ~R cer 6548 Rcnr 6549 0Rc0r 6550 +R cplr 6553 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-coll 3901 ax-sep 3904 ax-nul 3912 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-iinf 4337 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-csb 2910 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3259 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-iun 3688 df-br 3794 df-opab 3848 df-mpt 3849 df-tr 3884 df-eprel 4052 df-id 4056 df-po 4059 df-iso 4060 df-iord 4129 df-on 4131 df-suc 4134 df-iom 4340 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-f1 4937 df-fo 4938 df-f1o 4939 df-fv 4940 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-1st 5798 df-2nd 5799 df-recs 5954 df-irdg 6019 df-1o 6065 df-2o 6066 df-oadd 6069 df-omul 6070 df-er 6172 df-ec 6174 df-qs 6178 df-ni 6556 df-pli 6557 df-mi 6558 df-lti 6559 df-plpq 6596 df-mpq 6597 df-enq 6599 df-nqqs 6600 df-plqqs 6601 df-mqqs 6602 df-1nqqs 6603 df-rq 6604 df-ltnqqs 6605 df-enq0 6676 df-nq0 6677 df-0nq0 6678 df-plq0 6679 df-mq0 6680 df-inp 6718 df-i1p 6719 df-iplp 6720 df-enr 6965 df-nr 6966 df-plr 6967 df-0r 6970 |
This theorem is referenced by: addgt0sr 7014 ltadd1sr 7015 caucvgsrlemoffval 7034 caucvgsrlemoffres 7038 caucvgsr 7040 addresr 7067 mulresr 7068 axi2m1 7103 ax0id 7106 axcnre 7109 |
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