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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 6837 | . . . 4 ⊢ -1R ∈ R | |
2 | 1sr 6836 | . . . 4 ⊢ 1R ∈ R | |
3 | distrsrg 6844 | . . . 4 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R ∧ 1R ∈ R) → (𝐴 ·R (-1R +R 1R)) = ((𝐴 ·R -1R) +R (𝐴 ·R 1R))) | |
4 | 1, 2, 3 | mp3an23 1224 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R (-1R +R 1R)) = ((𝐴 ·R -1R) +R (𝐴 ·R 1R))) |
5 | m1p1sr 6845 | . . . . 5 ⊢ (-1R +R 1R) = 0R | |
6 | 5 | oveq2i 5523 | . . . 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 6839 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R) → (𝐴 ·R -1R) ∈ R) | |
9 | 1, 8 | mpan2 401 | . . . 4 ⊢ (𝐴 ∈ R → (𝐴 ·R -1R) ∈ R) |
10 | mulclsr 6839 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ 1R ∈ R) → (𝐴 ·R 1R) ∈ R) | |
11 | 2, 10 | mpan2 401 | . . . 4 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) ∈ R) |
12 | addcomsrg 6840 | . . . 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 391 | . . 3 ⊢ (𝐴 ∈ R → ((𝐴 ·R -1R) +R (𝐴 ·R 1R)) = ((𝐴 ·R 1R) +R (𝐴 ·R -1R))) |
14 | 4, 7, 13 | 3eqtr3d 2080 | . 2 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = ((𝐴 ·R 1R) +R (𝐴 ·R -1R))) |
15 | 00sr 6854 | . 2 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | |
16 | 1idsr 6853 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴) | |
17 | 16 | oveq1d 5527 | . 2 ⊢ (𝐴 ∈ R → ((𝐴 ·R 1R) +R (𝐴 ·R -1R)) = (𝐴 +R (𝐴 ·R -1R))) |
18 | 14, 15, 17 | 3eqtr3rd 2081 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 (class class class)co 5512 Rcnr 6395 0Rc0r 6396 1Rc1r 6397 -1Rcm1r 6398 +R cplr 6399 ·R cmr 6400 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-imp 6567 df-enr 6811 df-nr 6812 df-plr 6813 df-mr 6814 df-0r 6816 df-1r 6817 df-m1r 6818 |
This theorem is referenced by: negexsr 6857 caucvgsrlemoffval 6880 axrnegex 6953 |
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