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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 10366 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴) | |
2 | 1 | oveq1d 7031 | . 2 ⊢ (𝐴 ∈ R → ((𝐴 ·R 1R) +R (𝐴 ·R -1R)) = (𝐴 +R (𝐴 ·R -1R))) |
3 | distrsr 10359 | . . . 4 ⊢ (𝐴 ·R (-1R +R 1R)) = ((𝐴 ·R -1R) +R (𝐴 ·R 1R)) | |
4 | m1p1sr 10360 | . . . . 5 ⊢ (-1R +R 1R) = 0R | |
5 | 4 | oveq2i 7027 | . . . 4 ⊢ (𝐴 ·R (-1R +R 1R)) = (𝐴 ·R 0R) |
6 | addcomsr 10355 | . . . 4 ⊢ ((𝐴 ·R -1R) +R (𝐴 ·R 1R)) = ((𝐴 ·R 1R) +R (𝐴 ·R -1R)) | |
7 | 3, 5, 6 | 3eqtr3i 2827 | . . 3 ⊢ (𝐴 ·R 0R) = ((𝐴 ·R 1R) +R (𝐴 ·R -1R)) |
8 | 00sr 10367 | . . 3 ⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | |
9 | 7, 8 | syl5eqr 2845 | . 2 ⊢ (𝐴 ∈ R → ((𝐴 ·R 1R) +R (𝐴 ·R -1R)) = 0R) |
10 | 2, 9 | eqtr3d 2833 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 (class class class)co 7016 Rcnr 10133 0Rc0r 10134 1Rc1r 10135 -1Rcm1r 10136 +R cplr 10137 ·R cmr 10138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-omul 7958 df-er 8139 df-ec 8141 df-qs 8145 df-ni 10140 df-pli 10141 df-mi 10142 df-lti 10143 df-plpq 10176 df-mpq 10177 df-ltpq 10178 df-enq 10179 df-nq 10180 df-erq 10181 df-plq 10182 df-mq 10183 df-1nq 10184 df-rq 10185 df-ltnq 10186 df-np 10249 df-1p 10250 df-plp 10251 df-mp 10252 df-ltp 10253 df-enr 10323 df-nr 10324 df-plr 10325 df-mr 10326 df-0r 10328 df-1r 10329 df-m1r 10330 |
This theorem is referenced by: negexsr 10370 sqgt0sr 10374 map2psrpr 10378 axrnegex 10430 |
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