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Mirrors > Home > MPE Home > Th. List > Mathboxes > resubeulem2 | Structured version Visualization version GIF version |
Description: Lemma for resubeu 39256. 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 39252 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (0 −ℝ 𝐴)) = 0) |
3 | 2 | oveq1d 7171 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + (0 −ℝ 𝐴)) + ((0 −ℝ (0 + 0)) + 𝐵)) = (0 + ((0 −ℝ (0 + 0)) + 𝐵))) |
4 | simpl 485 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
5 | 4 | recnd 10669 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ) |
6 | rernegcl 39250 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 −ℝ 𝐴) ∈ ℝ) |
8 | 7 | recnd 10669 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 −ℝ 𝐴) ∈ ℂ) |
9 | elre0re 39203 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℝ) | |
10 | 9, 9 | readdcld 10670 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → (0 + 0) ∈ ℝ) |
11 | rernegcl 39250 | . . . . . . 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 10670 | . . . . 5 ⊢ (𝐵 ∈ ℝ → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℝ) |
15 | 14 | adantl 484 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℝ) |
16 | 15 | recnd 10669 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℂ) |
17 | 5, 8, 16 | addassd 10663 | . 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 39254 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (0 + (0 −ℝ (0 + 0))) = (0 −ℝ 0)) | |
19 | 18 | oveq1d 7171 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((0 + (0 −ℝ (0 + 0))) + 𝐵) = ((0 −ℝ 0) + 𝐵)) |
20 | 9 | recnd 10669 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℂ) |
21 | 12 | recnd 10669 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (0 −ℝ (0 + 0)) ∈ ℂ) |
22 | recn 10627 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
23 | 20, 21, 22 | addassd 10663 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((0 + (0 −ℝ (0 + 0))) + 𝐵) = (0 + ((0 −ℝ (0 + 0)) + 𝐵))) |
24 | reneg0addid2 39253 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((0 −ℝ 0) + 𝐵) = 𝐵) | |
25 | 19, 23, 24 | 3eqtr3d 2864 | . . 3 ⊢ (𝐵 ∈ ℝ → (0 + ((0 −ℝ (0 + 0)) + 𝐵)) = 𝐵) |
26 | 25 | adantl 484 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 + ((0 −ℝ (0 + 0)) + 𝐵)) = 𝐵) |
27 | 3, 17, 26 | 3eqtr3d 2864 | 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 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℝcr 10536 0cc0 10537 + caddc 10540 −ℝ cresub 39244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-addrcl 10598 ax-addass 10602 ax-rnegex 10608 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-resub 39245 |
This theorem is referenced by: resubeu 39256 |
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