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Mirrors > Home > MPE Home > Th. List > Mathboxes > renegid2 | Structured version Visualization version GIF version |
Description: Commuted version of renegid 39838. (Contributed by SN, 4-May-2024.) |
Ref | Expression |
---|---|
renegid2 | ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegid 39838 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0) | |
2 | 1 | oveq2d 7159 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + (𝐴 + (0 −ℝ 𝐴))) = ((0 −ℝ 𝐴) + 0)) |
3 | rernegcl 39836 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) | |
4 | readdid1 39874 | . . . . 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 2794 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + (𝐴 + (0 −ℝ 𝐴))) = (0 −ℝ 𝐴)) |
7 | 3 | recnd 10692 | . . . 4 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℂ) |
8 | recn 10650 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 7, 8, 7 | addassd 10686 | . . 3 ⊢ (𝐴 ∈ ℝ → (((0 −ℝ 𝐴) + 𝐴) + (0 −ℝ 𝐴)) = ((0 −ℝ 𝐴) + (𝐴 + (0 −ℝ 𝐴)))) |
10 | readdid2 39868 | . . . 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 2804 | . 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 10693 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) ∈ ℝ) |
15 | elre0re 39778 | . . 3 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
16 | readdcan2 39877 | . . 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 1369 | . 2 ⊢ (𝐴 ∈ ℝ → ((((0 −ℝ 𝐴) + 𝐴) + (0 −ℝ 𝐴)) = (0 + (0 −ℝ 𝐴)) ↔ ((0 −ℝ 𝐴) + 𝐴) = 0)) |
18 | 12, 17 | mpbid 235 | 1 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1539 ∈ wcel 2112 (class class class)co 7143 ℝcr 10559 0cc0 10560 + caddc 10563 −ℝ cresub 39830 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-po 5436 df-so 5437 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-ltxr 10703 df-2 11722 df-3 11723 df-resub 39831 |
This theorem is referenced by: sn-it0e0 39879 sn-negex12 39880 reixi 39886 sn-0tie0 39903 cnreeu 39920 |
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