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Mirrors > Home > MPE Home > Th. List > Mathboxes > readdid1addid2d | Structured version Visualization version GIF version |
Description: Given some real number 𝐵 where 𝐴 acts like a right additive identity, derive that 𝐴 is a left additive identity. Note that the hypothesis is weaker than proving that 𝐴 is a right additive identity (for all numbers). Although, if there is a right additive identity, then by readdcan 10800, 𝐴 is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.) |
Ref | Expression |
---|---|
readdid1addid2d.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
readdid1addid2d.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
readdid1addid2d.1 | ⊢ (𝜑 → (𝐵 + 𝐴) = 𝐵) |
Ref | Expression |
---|---|
readdid1addid2d | ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | readdid1addid2d.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | 1 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ) |
3 | 2 | recnd 10655 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℂ) |
4 | readdid1addid2d.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
5 | 4 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ) |
6 | 5 | recnd 10655 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℂ) |
7 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ) | |
8 | 7 | recnd 10655 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℂ) |
9 | 3, 6, 8 | addassd 10649 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + (𝐴 + 𝐶))) |
10 | readdid1addid2d.1 | . . . . 5 ⊢ (𝜑 → (𝐵 + 𝐴) = 𝐵) | |
11 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐵 + 𝐴) = 𝐵) |
12 | 11 | oveq1d 7157 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + 𝐶)) |
13 | 9, 12 | eqtr3d 2858 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶)) |
14 | 5, 7 | readdcld 10656 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) ∈ ℝ) |
15 | readdcan 10800 | . . 3 ⊢ (((𝐴 + 𝐶) ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶)) | |
16 | 14, 7, 2, 15 | syl3anc 1367 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶)) |
17 | 13, 16 | mpbid 234 | 1 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7142 ℝcr 10522 + caddc 10526 |
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 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-resscn 10580 ax-addrcl 10584 ax-addass 10588 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 |
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-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-po 5460 df-so 5461 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-ltxr 10666 |
This theorem is referenced by: reneg0addid2 39279 |
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