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Mirrors > Home > MPE Home > Th. List > Mathboxes > renegid2 | Structured version Visualization version GIF version |
Description: Commuted version of renegid 42351. (Contributed by SN, 4-May-2024.) |
Ref | Expression |
---|---|
renegid2 | ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegid 42351 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0) | |
2 | 1 | oveq2d 7466 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + (𝐴 + (0 −ℝ 𝐴))) = ((0 −ℝ 𝐴) + 0)) |
3 | rernegcl 42349 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) | |
4 | readdrid 42387 | . . . . 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 2780 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + (𝐴 + (0 −ℝ 𝐴))) = (0 −ℝ 𝐴)) |
7 | 3 | recnd 11320 | . . . 4 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℂ) |
8 | recn 11276 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 7, 8, 7 | addassd 11314 | . . 3 ⊢ (𝐴 ∈ ℝ → (((0 −ℝ 𝐴) + 𝐴) + (0 −ℝ 𝐴)) = ((0 −ℝ 𝐴) + (𝐴 + (0 −ℝ 𝐴)))) |
10 | readdlid 42381 | . . . 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 2790 | . 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 11321 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) ∈ ℝ) |
15 | elre0re 42251 | . . 3 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
16 | readdcan2 42390 | . . 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 1371 | . 2 ⊢ (𝐴 ∈ ℝ → ((((0 −ℝ 𝐴) + 𝐴) + (0 −ℝ 𝐴)) = (0 + (0 −ℝ 𝐴)) ↔ ((0 −ℝ 𝐴) + 𝐴) = 0)) |
18 | 12, 17 | mpbid 232 | 1 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2108 (class class class)co 7450 ℝcr 11185 0cc0 11186 + caddc 11189 −ℝ cresub 42343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-ltxr 11331 df-2 12358 df-3 12359 df-resub 42344 |
This theorem is referenced by: sn-it0e0 42393 sn-negex12 42394 reixi 42400 sn-0tie0 42417 zaddcomlem 42429 zaddcom 42430 cnreeu 42448 |
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