Theorem readdridaddlidd 42748

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 11318, 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ)
3	2	recnd 11171	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℂ)
4		readdridaddlidd.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ)
6	5	recnd 11171	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℂ)
7		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ)
8	7	recnd 11171	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℂ)
9	3, 6, 8	addassd 11165	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + (𝐴 + 𝐶)))
10		readdridaddlidd.1	. . . . 5 ⊢ (𝜑 → (𝐵 + 𝐴) = 𝐵)
11	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐵 + 𝐴) = 𝐵)
12	11	oveq1d 7378	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + 𝐶))
13	9, 12	eqtr3d 2777	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶))
14	5, 7	readdcld 11172	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) ∈ ℝ)
15		readdcan 11318	. . 3 ⊢ (((𝐴 + 𝐶) ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶))
16	14, 7, 2, 15	syl3anc 1379	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶))
17	13, 16	mpbid 233	1 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  (class class class)co 7363  ℝcr 11035   + caddc 11039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-resscn 11093  ax-addrcl 11097  ax-addass 11101  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-ltxr 11182

This theorem is referenced by:  reneg0addlid  42858