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Mirrors > Home > MPE Home > Th. List > Mathboxes > resubeulem2 | Structured version Visualization version GIF version |
Description: Lemma for resubeu 40068. A value which when added to 𝐴, results in 𝐵. (Contributed by Steven Nguyen, 7-Jan-2023.) |
Ref | Expression |
---|---|
resubeulem2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + ((0 −ℝ 𝐴) + ((0 −ℝ (0 + 0)) + 𝐵))) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegid 40064 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0) | |
2 | 1 | adantr 484 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (0 −ℝ 𝐴)) = 0) |
3 | 2 | oveq1d 7228 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + (0 −ℝ 𝐴)) + ((0 −ℝ (0 + 0)) + 𝐵)) = (0 + ((0 −ℝ (0 + 0)) + 𝐵))) |
4 | simpl 486 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
5 | 4 | recnd 10861 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ) |
6 | rernegcl 40062 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) | |
7 | 6 | adantr 484 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 −ℝ 𝐴) ∈ ℝ) |
8 | 7 | recnd 10861 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 −ℝ 𝐴) ∈ ℂ) |
9 | elre0re 39998 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℝ) | |
10 | 9, 9 | readdcld 10862 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → (0 + 0) ∈ ℝ) |
11 | rernegcl 40062 | . . . . . . 7 ⊢ ((0 + 0) ∈ ℝ → (0 −ℝ (0 + 0)) ∈ ℝ) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → (0 −ℝ (0 + 0)) ∈ ℝ) |
13 | id 22 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ) | |
14 | 12, 13 | readdcld 10862 | . . . . 5 ⊢ (𝐵 ∈ ℝ → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℝ) |
15 | 14 | adantl 485 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℝ) |
16 | 15 | recnd 10861 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℂ) |
17 | 5, 8, 16 | addassd 10855 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + (0 −ℝ 𝐴)) + ((0 −ℝ (0 + 0)) + 𝐵)) = (𝐴 + ((0 −ℝ 𝐴) + ((0 −ℝ (0 + 0)) + 𝐵)))) |
18 | resubeulem1 40066 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (0 + (0 −ℝ (0 + 0))) = (0 −ℝ 0)) | |
19 | 18 | oveq1d 7228 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((0 + (0 −ℝ (0 + 0))) + 𝐵) = ((0 −ℝ 0) + 𝐵)) |
20 | 9 | recnd 10861 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℂ) |
21 | 12 | recnd 10861 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (0 −ℝ (0 + 0)) ∈ ℂ) |
22 | recn 10819 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
23 | 20, 21, 22 | addassd 10855 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((0 + (0 −ℝ (0 + 0))) + 𝐵) = (0 + ((0 −ℝ (0 + 0)) + 𝐵))) |
24 | reneg0addid2 40065 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((0 −ℝ 0) + 𝐵) = 𝐵) | |
25 | 19, 23, 24 | 3eqtr3d 2785 | . . 3 ⊢ (𝐵 ∈ ℝ → (0 + ((0 −ℝ (0 + 0)) + 𝐵)) = 𝐵) |
26 | 25 | adantl 485 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 + ((0 −ℝ (0 + 0)) + 𝐵)) = 𝐵) |
27 | 3, 17, 26 | 3eqtr3d 2785 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + ((0 −ℝ 𝐴) + ((0 −ℝ (0 + 0)) + 𝐵))) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 (class class class)co 7213 ℝcr 10728 0cc0 10729 + caddc 10732 −ℝ cresub 40056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-addrcl 10790 ax-addass 10794 ax-rnegex 10800 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-ltxr 10872 df-resub 40057 |
This theorem is referenced by: resubeu 40068 |
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