Theorem readdridaddlidd 42696

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 11320, 𝐴 is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.)

Hypotheses

Ref	Expression
readdridaddlidd.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
readdridaddlidd.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
readdridaddlidd.1	⊢ (𝜑 → (𝐵 + 𝐴) = 𝐵)

Assertion

Ref	Expression
readdridaddlidd	⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶)

Proof of Theorem readdridaddlidd

Step	Hyp	Ref	Expression
1		readdridaddlidd.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
2	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ)
3	2	recnd 11173	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℂ)
4		readdridaddlidd.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ)
6	5	recnd 11173	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℂ)
7		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ)
8	7	recnd 11173	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℂ)
9	3, 6, 8	addassd 11167	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + (𝐴 + 𝐶)))
10		readdridaddlidd.1	. . . . 5 ⊢ (𝜑 → (𝐵 + 𝐴) = 𝐵)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐵 + 𝐴) = 𝐵)
12	11	oveq1d 7382	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + 𝐶))
13	9, 12	eqtr3d 2773	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶))
14	5, 7	readdcld 11174	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) ∈ ℝ)
15		readdcan 11320	. . 3 ⊢ (((𝐴 + 𝐶) ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶))
16	14, 7, 2, 15	syl3anc 1374	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶))
17	13, 16	mpbid 232	1 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  (class class class)co 7367  ℝcr 11037   + caddc 11041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-resscn 11095  ax-addrcl 11099  ax-addass 11103  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-ltxr 11184

This theorem is referenced by:  reneg0addlid  42806