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Mirrors > Home > MPE Home > Th. List > Mathboxes > renegid2 | Structured version Visualization version GIF version |
Description: Commuted version of renegid 42282. (Contributed by SN, 4-May-2024.) |
Ref | Expression |
---|---|
renegid2 | ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegid 42282 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0) | |
2 | 1 | oveq2d 7461 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + (𝐴 + (0 −ℝ 𝐴))) = ((0 −ℝ 𝐴) + 0)) |
3 | rernegcl 42280 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) | |
4 | readdrid 42318 | . . . . 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 2774 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + (𝐴 + (0 −ℝ 𝐴))) = (0 −ℝ 𝐴)) |
7 | 3 | recnd 11314 | . . . 4 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℂ) |
8 | recn 11270 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 7, 8, 7 | addassd 11308 | . . 3 ⊢ (𝐴 ∈ ℝ → (((0 −ℝ 𝐴) + 𝐴) + (0 −ℝ 𝐴)) = ((0 −ℝ 𝐴) + (𝐴 + (0 −ℝ 𝐴)))) |
10 | readdlid 42312 | . . . 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 2784 | . 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 11315 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) ∈ ℝ) |
15 | elre0re 42198 | . . 3 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
16 | readdcan2 42321 | . . 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 2103 (class class class)co 7445 ℝcr 11179 0cc0 11180 + caddc 11183 −ℝ cresub 42274 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-po 5611 df-so 5612 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-ltxr 11325 df-2 12352 df-3 12353 df-resub 42275 |
This theorem is referenced by: sn-it0e0 42324 sn-negex12 42325 reixi 42331 sn-0tie0 42348 zaddcomlem 42360 zaddcom 42361 cnreeu 42379 |
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