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Mirrors > Home > MPE Home > Th. List > Mathboxes > resubeulem1 | Structured version Visualization version GIF version |
Description: Lemma for resubeu 39515. A value which when added to zero, results in negative zero. (Contributed by Steven Nguyen, 7-Jan-2023.) |
Ref | Expression |
---|---|
resubeulem1 | ⊢ (𝐴 ∈ ℝ → (0 + (0 −ℝ (0 + 0))) = (0 −ℝ 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elre0re 39462 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
2 | 1 | recnd 10658 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℂ) |
3 | 1, 1 | readdcld 10659 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (0 + 0) ∈ ℝ) |
4 | rernegcl 39509 | . . . . . . 7 ⊢ ((0 + 0) ∈ ℝ → (0 −ℝ (0 + 0)) ∈ ℝ) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 −ℝ (0 + 0)) ∈ ℝ) |
6 | 5 | recnd 10658 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 −ℝ (0 + 0)) ∈ ℂ) |
7 | 2, 2, 6 | addassd 10652 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0 + 0) + (0 −ℝ (0 + 0))) = (0 + (0 + (0 −ℝ (0 + 0))))) |
8 | renegid 39511 | . . . . 5 ⊢ ((0 + 0) ∈ ℝ → ((0 + 0) + (0 −ℝ (0 + 0))) = 0) | |
9 | 3, 8 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0 + 0) + (0 −ℝ (0 + 0))) = 0) |
10 | 7, 9 | eqtr3d 2835 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 + (0 + (0 −ℝ (0 + 0)))) = 0) |
11 | 1, 5 | readdcld 10659 | . . . 4 ⊢ (𝐴 ∈ ℝ → (0 + (0 −ℝ (0 + 0))) ∈ ℝ) |
12 | renegadd 39510 | . . . 4 ⊢ ((0 ∈ ℝ ∧ (0 + (0 −ℝ (0 + 0))) ∈ ℝ) → ((0 −ℝ 0) = (0 + (0 −ℝ (0 + 0))) ↔ (0 + (0 + (0 −ℝ (0 + 0)))) = 0)) | |
13 | 1, 11, 12 | syl2anc 587 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 0) = (0 + (0 −ℝ (0 + 0))) ↔ (0 + (0 + (0 −ℝ (0 + 0)))) = 0)) |
14 | 10, 13 | mpbird 260 | . 2 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 0) = (0 + (0 −ℝ (0 + 0)))) |
15 | 14 | eqcomd 2804 | 1 ⊢ (𝐴 ∈ ℝ → (0 + (0 −ℝ (0 + 0))) = (0 −ℝ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℝcr 10525 0cc0 10526 + caddc 10529 −ℝ cresub 39503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-addrcl 10587 ax-addass 10591 ax-rnegex 10597 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-resub 39504 |
This theorem is referenced by: resubeulem2 39514 |
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