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Mirrors > Home > MPE Home > Th. List > Mathboxes > renegid2 | Structured version Visualization version GIF version |
Description: Commuted version of renegid 41190. (Contributed by SN, 4-May-2024.) |
Ref | Expression |
---|---|
renegid2 | ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegid 41190 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0) | |
2 | 1 | oveq2d 7420 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + (𝐴 + (0 −ℝ 𝐴))) = ((0 −ℝ 𝐴) + 0)) |
3 | rernegcl 41188 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) | |
4 | readdrid 41226 | . . . . 5 ⊢ ((0 −ℝ 𝐴) ∈ ℝ → ((0 −ℝ 𝐴) + 0) = (0 −ℝ 𝐴)) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 0) = (0 −ℝ 𝐴)) |
6 | 2, 5 | eqtrd 2773 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + (𝐴 + (0 −ℝ 𝐴))) = (0 −ℝ 𝐴)) |
7 | 3 | recnd 11238 | . . . 4 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℂ) |
8 | recn 11196 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 7, 8, 7 | addassd 11232 | . . 3 ⊢ (𝐴 ∈ ℝ → (((0 −ℝ 𝐴) + 𝐴) + (0 −ℝ 𝐴)) = ((0 −ℝ 𝐴) + (𝐴 + (0 −ℝ 𝐴)))) |
10 | readdlid 41220 | . . . 4 ⊢ ((0 −ℝ 𝐴) ∈ ℝ → (0 + (0 −ℝ 𝐴)) = (0 −ℝ 𝐴)) | |
11 | 3, 10 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 + (0 −ℝ 𝐴)) = (0 −ℝ 𝐴)) |
12 | 6, 9, 11 | 3eqtr4d 2783 | . 2 ⊢ (𝐴 ∈ ℝ → (((0 −ℝ 𝐴) + 𝐴) + (0 −ℝ 𝐴)) = (0 + (0 −ℝ 𝐴))) |
13 | id 22 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ) | |
14 | 3, 13 | readdcld 11239 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) ∈ ℝ) |
15 | elre0re 41125 | . . 3 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
16 | readdcan2 41229 | . . 3 ⊢ ((((0 −ℝ 𝐴) + 𝐴) ∈ ℝ ∧ 0 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℝ) → ((((0 −ℝ 𝐴) + 𝐴) + (0 −ℝ 𝐴)) = (0 + (0 −ℝ 𝐴)) ↔ ((0 −ℝ 𝐴) + 𝐴) = 0)) | |
17 | 14, 15, 3, 16 | syl3anc 1372 | . 2 ⊢ (𝐴 ∈ ℝ → ((((0 −ℝ 𝐴) + 𝐴) + (0 −ℝ 𝐴)) = (0 + (0 −ℝ 𝐴)) ↔ ((0 −ℝ 𝐴) + 𝐴) = 0)) |
18 | 12, 17 | mpbid 231 | 1 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 (class class class)co 7404 ℝcr 11105 0cc0 11106 + caddc 11109 −ℝ cresub 41182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-2 12271 df-3 12272 df-resub 41183 |
This theorem is referenced by: sn-it0e0 41232 sn-negex12 41233 reixi 41239 sn-0tie0 41256 zaddcomlem 41268 zaddcom 41269 cnreeu 41285 |
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