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Mirrors > Home > MPE Home > Th. List > Mathboxes > resubeulem2 | Structured version Visualization version GIF version |
Description: Lemma for resubeu 42384. 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 42380 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (0 −ℝ 𝐴)) = 0) |
3 | 2 | oveq1d 7446 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + (0 −ℝ 𝐴)) + ((0 −ℝ (0 + 0)) + 𝐵)) = (0 + ((0 −ℝ (0 + 0)) + 𝐵))) |
4 | simpl 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
5 | 4 | recnd 11287 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ) |
6 | rernegcl 42378 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) | |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 −ℝ 𝐴) ∈ ℝ) |
8 | 7 | recnd 11287 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 −ℝ 𝐴) ∈ ℂ) |
9 | elre0re 42274 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℝ) | |
10 | 9, 9 | readdcld 11288 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → (0 + 0) ∈ ℝ) |
11 | rernegcl 42378 | . . . . . . 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 11288 | . . . . 5 ⊢ (𝐵 ∈ ℝ → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℝ) |
15 | 14 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℝ) |
16 | 15 | recnd 11287 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℂ) |
17 | 5, 8, 16 | addassd 11281 | . 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 42382 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (0 + (0 −ℝ (0 + 0))) = (0 −ℝ 0)) | |
19 | 18 | oveq1d 7446 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((0 + (0 −ℝ (0 + 0))) + 𝐵) = ((0 −ℝ 0) + 𝐵)) |
20 | 9 | recnd 11287 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℂ) |
21 | 12 | recnd 11287 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (0 −ℝ (0 + 0)) ∈ ℂ) |
22 | recn 11243 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
23 | 20, 21, 22 | addassd 11281 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((0 + (0 −ℝ (0 + 0))) + 𝐵) = (0 + ((0 −ℝ (0 + 0)) + 𝐵))) |
24 | reneg0addlid 42381 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((0 −ℝ 0) + 𝐵) = 𝐵) | |
25 | 19, 23, 24 | 3eqtr3d 2783 | . . 3 ⊢ (𝐵 ∈ ℝ → (0 + ((0 −ℝ (0 + 0)) + 𝐵)) = 𝐵) |
26 | 25 | adantl 481 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 + ((0 −ℝ (0 + 0)) + 𝐵)) = 𝐵) |
27 | 3, 17, 26 | 3eqtr3d 2783 | 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 395 = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℝcr 11152 0cc0 11153 + caddc 11156 −ℝ cresub 42372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-addrcl 11214 ax-addass 11218 ax-rnegex 11224 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-resub 42373 |
This theorem is referenced by: resubeu 42384 |
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