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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 7201 | . . . 4 ⊢ -1R ∈ R | |
2 | 1sr 7200 | . . . 4 ⊢ 1R ∈ R | |
3 | distrsrg 7208 | . . . 4 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R ∧ 1R ∈ R) → (𝐴 ·R (-1R +R 1R)) = ((𝐴 ·R -1R) +R (𝐴 ·R 1R))) | |
4 | 1, 2, 3 | mp3an23 1261 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R (-1R +R 1R)) = ((𝐴 ·R -1R) +R (𝐴 ·R 1R))) |
5 | m1p1sr 7209 | . . . . 5 ⊢ (-1R +R 1R) = 0R | |
6 | 5 | oveq2i 5602 | . . . 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 7203 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R) → (𝐴 ·R -1R) ∈ R) | |
9 | 1, 8 | mpan2 416 | . . . 4 ⊢ (𝐴 ∈ R → (𝐴 ·R -1R) ∈ R) |
10 | mulclsr 7203 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ 1R ∈ R) → (𝐴 ·R 1R) ∈ R) | |
11 | 2, 10 | mpan2 416 | . . . 4 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) ∈ R) |
12 | addcomsrg 7204 | . . . 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 2123 | . 2 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = ((𝐴 ·R 1R) +R (𝐴 ·R -1R))) |
15 | 00sr 7218 | . 2 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | |
16 | 1idsr 7217 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴) | |
17 | 16 | oveq1d 5606 | . 2 ⊢ (𝐴 ∈ R → ((𝐴 ·R 1R) +R (𝐴 ·R -1R)) = (𝐴 +R (𝐴 ·R -1R))) |
18 | 14, 15, 17 | 3eqtr3rd 2124 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 (class class class)co 5591 Rcnr 6759 0Rc0r 6760 1Rc1r 6761 -1Rcm1r 6762 +R cplr 6763 ·R cmr 6764 |
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 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 |
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 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-eprel 4080 df-id 4084 df-po 4087 df-iso 4088 df-iord 4157 df-on 4159 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-recs 6002 df-irdg 6067 df-1o 6113 df-2o 6114 df-oadd 6117 df-omul 6118 df-er 6222 df-ec 6224 df-qs 6228 df-ni 6766 df-pli 6767 df-mi 6768 df-lti 6769 df-plpq 6806 df-mpq 6807 df-enq 6809 df-nqqs 6810 df-plqqs 6811 df-mqqs 6812 df-1nqqs 6813 df-rq 6814 df-ltnqqs 6815 df-enq0 6886 df-nq0 6887 df-0nq0 6888 df-plq0 6889 df-mq0 6890 df-inp 6928 df-i1p 6929 df-iplp 6930 df-imp 6931 df-enr 7175 df-nr 7176 df-plr 7177 df-mr 7178 df-0r 7180 df-1r 7181 df-m1r 7182 |
This theorem is referenced by: negexsr 7221 caucvgsrlemoffval 7244 axrnegex 7317 |
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