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Mirrors > Home > MPE Home > Th. List > Mathboxes > renegid | Structured version Visualization version GIF version |
Description: Addition of a real number and its negative. (Contributed by Steven Nguyen, 7-Jan-2023.) |
Ref | Expression |
---|---|
renegid | ⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . 2 ⊢ (0 −ℝ 𝐴) = (0 −ℝ 𝐴) | |
2 | rernegcl 38633 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) | |
3 | renegadd 38634 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℝ) → ((0 −ℝ 𝐴) = (0 −ℝ 𝐴) ↔ (𝐴 + (0 −ℝ 𝐴)) = 0)) | |
4 | 2, 3 | mpdan 674 | . 2 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) = (0 −ℝ 𝐴) ↔ (𝐴 + (0 −ℝ 𝐴)) = 0)) |
5 | 1, 4 | mpbii 225 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2050 (class class class)co 6978 ℝcr 10336 0cc0 10337 + caddc 10340 −ℝ cresub 38627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-resscn 10394 ax-addrcl 10398 ax-rnegex 10408 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-opab 4993 df-mpt 5010 df-id 5313 df-po 5327 df-so 5328 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-pnf 10478 df-mnf 10479 df-ltxr 10481 df-resub 38628 |
This theorem is referenced by: reneg0addid1 38637 resubeulem1 38638 resubeulem2 38639 |
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