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Mirrors > Home > MPE Home > Th. List > pn0sr | Structured version Visualization version GIF version |
Description: A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pn0sr | ⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1idsr 10508 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴) | |
2 | 1 | oveq1d 7160 | . 2 ⊢ (𝐴 ∈ R → ((𝐴 ·R 1R) +R (𝐴 ·R -1R)) = (𝐴 +R (𝐴 ·R -1R))) |
3 | distrsr 10501 | . . . 4 ⊢ (𝐴 ·R (-1R +R 1R)) = ((𝐴 ·R -1R) +R (𝐴 ·R 1R)) | |
4 | m1p1sr 10502 | . . . . 5 ⊢ (-1R +R 1R) = 0R | |
5 | 4 | oveq2i 7156 | . . . 4 ⊢ (𝐴 ·R (-1R +R 1R)) = (𝐴 ·R 0R) |
6 | addcomsr 10497 | . . . 4 ⊢ ((𝐴 ·R -1R) +R (𝐴 ·R 1R)) = ((𝐴 ·R 1R) +R (𝐴 ·R -1R)) | |
7 | 3, 5, 6 | 3eqtr3i 2849 | . . 3 ⊢ (𝐴 ·R 0R) = ((𝐴 ·R 1R) +R (𝐴 ·R -1R)) |
8 | 00sr 10509 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | |
9 | 7, 8 | syl5eqr 2867 | . 2 ⊢ (𝐴 ∈ R → ((𝐴 ·R 1R) +R (𝐴 ·R -1R)) = 0R) |
10 | 2, 9 | eqtr3d 2855 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 Rcnr 10275 0Rc0r 10276 1Rc1r 10277 -1Rcm1r 10278 +R cplr 10279 ·R cmr 10280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-omul 8096 df-er 8278 df-ec 8280 df-qs 8284 df-ni 10282 df-pli 10283 df-mi 10284 df-lti 10285 df-plpq 10318 df-mpq 10319 df-ltpq 10320 df-enq 10321 df-nq 10322 df-erq 10323 df-plq 10324 df-mq 10325 df-1nq 10326 df-rq 10327 df-ltnq 10328 df-np 10391 df-1p 10392 df-plp 10393 df-mp 10394 df-ltp 10395 df-enr 10465 df-nr 10466 df-plr 10467 df-mr 10468 df-0r 10470 df-1r 10471 df-m1r 10472 |
This theorem is referenced by: negexsr 10512 sqgt0sr 10516 map2psrpr 10520 axrnegex 10572 |
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