Theorem readdridaddlidd 42376

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 11294, 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 11147	. . . 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 11147	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℂ)
7		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ)
8	7	recnd 11147	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℂ)
9	3, 6, 8	addassd 11141	. . 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 7367	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + 𝐶))
13	9, 12	eqtr3d 2770	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶))
14	5, 7	readdcld 11148	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) ∈ ℝ)
15		readdcan 11294	. . 3 ⊢ (((𝐴 + 𝐶) ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶))
16	14, 7, 2, 15	syl3anc 1373	. 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 1541   ∈ wcel 2113  (class class class)co 7352  ℝcr 11012   + caddc 11016

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-resscn 11070  ax-addrcl 11074  ax-addass 11078  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-pnf 11155  df-mnf 11156  df-ltxr 11158

This theorem is referenced by:  reneg0addlid  42492