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Mirrors > Home > ILE Home > Th. List > pn0sr | GIF version |
Description: A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) |
Ref | Expression |
---|---|
pn0sr | ⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | m1r 7296 | . . . 4 ⊢ -1R ∈ R | |
2 | 1sr 7295 | . . . 4 ⊢ 1R ∈ R | |
3 | distrsrg 7303 | . . . 4 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R ∧ 1R ∈ R) → (𝐴 ·R (-1R +R 1R)) = ((𝐴 ·R -1R) +R (𝐴 ·R 1R))) | |
4 | 1, 2, 3 | mp3an23 1265 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R (-1R +R 1R)) = ((𝐴 ·R -1R) +R (𝐴 ·R 1R))) |
5 | m1p1sr 7304 | . . . . 5 ⊢ (-1R +R 1R) = 0R | |
6 | 5 | oveq2i 5663 | . . . 4 ⊢ (𝐴 ·R (-1R +R 1R)) = (𝐴 ·R 0R) |
7 | 6 | a1i 9 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R (-1R +R 1R)) = (𝐴 ·R 0R)) |
8 | mulclsr 7298 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R) → (𝐴 ·R -1R) ∈ R) | |
9 | 1, 8 | mpan2 416 | . . . 4 ⊢ (𝐴 ∈ R → (𝐴 ·R -1R) ∈ R) |
10 | mulclsr 7298 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ 1R ∈ R) → (𝐴 ·R 1R) ∈ R) | |
11 | 2, 10 | mpan2 416 | . . . 4 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) ∈ R) |
12 | addcomsrg 7299 | . . . 4 ⊢ (((𝐴 ·R -1R) ∈ R ∧ (𝐴 ·R 1R) ∈ R) → ((𝐴 ·R -1R) +R (𝐴 ·R 1R)) = ((𝐴 ·R 1R) +R (𝐴 ·R -1R))) | |
13 | 9, 11, 12 | syl2anc 403 | . . 3 ⊢ (𝐴 ∈ R → ((𝐴 ·R -1R) +R (𝐴 ·R 1R)) = ((𝐴 ·R 1R) +R (𝐴 ·R -1R))) |
14 | 4, 7, 13 | 3eqtr3d 2128 | . 2 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = ((𝐴 ·R 1R) +R (𝐴 ·R -1R))) |
15 | 00sr 7313 | . 2 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | |
16 | 1idsr 7312 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴) | |
17 | 16 | oveq1d 5667 | . 2 ⊢ (𝐴 ∈ R → ((𝐴 ·R 1R) +R (𝐴 ·R -1R)) = (𝐴 +R (𝐴 ·R -1R))) |
18 | 14, 15, 17 | 3eqtr3rd 2129 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 ∈ wcel 1438 (class class class)co 5652 Rcnr 6854 0Rc0r 6855 1Rc1r 6856 -1Rcm1r 6857 +R cplr 6858 ·R cmr 6859 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-eprel 4116 df-id 4120 df-po 4123 df-iso 4124 df-iord 4193 df-on 4195 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-irdg 6135 df-1o 6181 df-2o 6182 df-oadd 6185 df-omul 6186 df-er 6290 df-ec 6292 df-qs 6296 df-ni 6861 df-pli 6862 df-mi 6863 df-lti 6864 df-plpq 6901 df-mpq 6902 df-enq 6904 df-nqqs 6905 df-plqqs 6906 df-mqqs 6907 df-1nqqs 6908 df-rq 6909 df-ltnqqs 6910 df-enq0 6981 df-nq0 6982 df-0nq0 6983 df-plq0 6984 df-mq0 6985 df-inp 7023 df-i1p 7024 df-iplp 7025 df-imp 7026 df-enr 7270 df-nr 7271 df-plr 7272 df-mr 7273 df-0r 7275 df-1r 7276 df-m1r 7277 |
This theorem is referenced by: negexsr 7316 caucvgsrlemoffval 7339 axrnegex 7412 |
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