![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > resubeulem1 | Structured version Visualization version GIF version |
Description: Lemma for resubeu 41844. 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 41748 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
2 | 1 | recnd 11258 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℂ) |
3 | 1, 1 | readdcld 11259 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (0 + 0) ∈ ℝ) |
4 | rernegcl 41838 | . . . . . . 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 11258 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 −ℝ (0 + 0)) ∈ ℂ) |
7 | 2, 2, 6 | addassd 11252 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0 + 0) + (0 −ℝ (0 + 0))) = (0 + (0 + (0 −ℝ (0 + 0))))) |
8 | renegid 41840 | . . . . 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 2769 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 + (0 + (0 −ℝ (0 + 0)))) = 0) |
11 | 1, 5 | readdcld 11259 | . . . 4 ⊢ (𝐴 ∈ ℝ → (0 + (0 −ℝ (0 + 0))) ∈ ℝ) |
12 | renegadd 41839 | . . . 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 583 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 0) = (0 + (0 −ℝ (0 + 0))) ↔ (0 + (0 + (0 −ℝ (0 + 0)))) = 0)) |
14 | 10, 13 | mpbird 257 | . 2 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 0) = (0 + (0 −ℝ (0 + 0)))) |
15 | 14 | eqcomd 2733 | 1 ⊢ (𝐴 ∈ ℝ → (0 + (0 −ℝ (0 + 0))) = (0 −ℝ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 (class class class)co 7414 ℝcr 11123 0cc0 11124 + caddc 11127 −ℝ cresub 41832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-resscn 11181 ax-addrcl 11185 ax-addass 11189 ax-rnegex 11195 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-ltxr 11269 df-resub 41833 |
This theorem is referenced by: resubeulem2 41843 |
Copyright terms: Public domain | W3C validator |