Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7673 | . . . 4 ⊢ -1R ∈ R | |
2 | 1sr 7672 | . . . 4 ⊢ 1R ∈ R | |
3 | distrsrg 7680 | . . . 4 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R ∧ 1R ∈ R) → (𝐴 ·R (-1R +R 1R)) = ((𝐴 ·R -1R) +R (𝐴 ·R 1R))) | |
4 | 1, 2, 3 | mp3an23 1311 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R (-1R +R 1R)) = ((𝐴 ·R -1R) +R (𝐴 ·R 1R))) |
5 | m1p1sr 7681 | . . . . 5 ⊢ (-1R +R 1R) = 0R | |
6 | 5 | oveq2i 5836 | . . . 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 7675 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ -1R ∈ R) → (𝐴 ·R -1R) ∈ R) | |
9 | 1, 8 | mpan2 422 | . . . 4 ⊢ (𝐴 ∈ R → (𝐴 ·R -1R) ∈ R) |
10 | mulclsr 7675 | . . . . 5 ⊢ ((𝐴 ∈ R ∧ 1R ∈ R) → (𝐴 ·R 1R) ∈ R) | |
11 | 2, 10 | mpan2 422 | . . . 4 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) ∈ R) |
12 | addcomsrg 7676 | . . . 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 409 | . . 3 ⊢ (𝐴 ∈ R → ((𝐴 ·R -1R) +R (𝐴 ·R 1R)) = ((𝐴 ·R 1R) +R (𝐴 ·R -1R))) |
14 | 4, 7, 13 | 3eqtr3d 2198 | . 2 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = ((𝐴 ·R 1R) +R (𝐴 ·R -1R))) |
15 | 00sr 7690 | . 2 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | |
16 | 1idsr 7689 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴) | |
17 | 16 | oveq1d 5840 | . 2 ⊢ (𝐴 ∈ R → ((𝐴 ·R 1R) +R (𝐴 ·R -1R)) = (𝐴 +R (𝐴 ·R -1R))) |
18 | 14, 15, 17 | 3eqtr3rd 2199 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∈ wcel 2128 (class class class)co 5825 Rcnr 7218 0Rc0r 7219 1Rc1r 7220 -1Rcm1r 7221 +R cplr 7222 ·R cmr 7223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-eprel 4250 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-irdg 6318 df-1o 6364 df-2o 6365 df-oadd 6368 df-omul 6369 df-er 6481 df-ec 6483 df-qs 6487 df-ni 7225 df-pli 7226 df-mi 7227 df-lti 7228 df-plpq 7265 df-mpq 7266 df-enq 7268 df-nqqs 7269 df-plqqs 7270 df-mqqs 7271 df-1nqqs 7272 df-rq 7273 df-ltnqqs 7274 df-enq0 7345 df-nq0 7346 df-0nq0 7347 df-plq0 7348 df-mq0 7349 df-inp 7387 df-i1p 7388 df-iplp 7389 df-imp 7390 df-enr 7647 df-nr 7648 df-plr 7649 df-mr 7650 df-0r 7652 df-1r 7653 df-m1r 7654 |
This theorem is referenced by: negexsr 7693 caucvgsrlemoffval 7717 map2psrprg 7726 axrnegex 7800 |
Copyright terms: Public domain | W3C validator |