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Mirrors > Home > MPE Home > Th. List > Mathboxes > resubeulem2 | Structured version Visualization version GIF version |
Description: Lemma for resubeu 38678. 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 38673 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0) | |
2 | 1 | adantr 473 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (0 −ℝ 𝐴)) = 0) |
3 | 2 | oveq1d 6989 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + (0 −ℝ 𝐴)) + ((0 −ℝ (0 + 0)) + 𝐵)) = (0 + ((0 −ℝ (0 + 0)) + 𝐵))) |
4 | simpl 475 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
5 | 4 | recnd 10466 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℂ) |
6 | rernegcl 38671 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) | |
7 | 6 | adantr 473 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 −ℝ 𝐴) ∈ ℝ) |
8 | 7 | recnd 10466 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 −ℝ 𝐴) ∈ ℂ) |
9 | elre0re 38629 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℝ) | |
10 | 9, 9 | readdcld 10467 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → (0 + 0) ∈ ℝ) |
11 | rernegcl 38671 | . . . . . . 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 10467 | . . . . 5 ⊢ (𝐵 ∈ ℝ → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℝ) |
15 | 14 | adantl 474 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℝ) |
16 | 15 | recnd 10466 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℂ) |
17 | 5, 8, 16 | addassd 10460 | . 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 38676 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (0 + (0 −ℝ (0 + 0))) = (0 −ℝ 0)) | |
19 | 18 | oveq1d 6989 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((0 + (0 −ℝ (0 + 0))) + 𝐵) = ((0 −ℝ 0) + 𝐵)) |
20 | 9 | recnd 10466 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℂ) |
21 | 12 | recnd 10466 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (0 −ℝ (0 + 0)) ∈ ℂ) |
22 | recn 10423 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
23 | 20, 21, 22 | addassd 10460 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((0 + (0 −ℝ (0 + 0))) + 𝐵) = (0 + ((0 −ℝ (0 + 0)) + 𝐵))) |
24 | reneg0addid1 38675 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((0 −ℝ 0) + 𝐵) = 𝐵) | |
25 | 19, 23, 24 | 3eqtr3d 2815 | . . 3 ⊢ (𝐵 ∈ ℝ → (0 + ((0 −ℝ (0 + 0)) + 𝐵)) = 𝐵) |
26 | 25 | adantl 474 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 + ((0 −ℝ (0 + 0)) + 𝐵)) = 𝐵) |
27 | 3, 17, 26 | 3eqtr3d 2815 | 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 387 = wceq 1508 ∈ wcel 2051 (class class class)co 6974 ℝcr 10332 0cc0 10333 + caddc 10336 −ℝ cresub 38665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-resscn 10390 ax-addrcl 10394 ax-addass 10398 ax-rnegex 10404 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-po 5322 df-so 5323 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-ltxr 10477 df-resub 38666 |
This theorem is referenced by: resubeu 38678 |
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