Theorem readdridaddlidd 43276

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 11465, 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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ)
3	2	recnd 11318	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℂ)
4		readdridaddlidd.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ)
6	5	recnd 11318	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℂ)
7		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ)
8	7	recnd 11318	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℂ)
9	3, 6, 8	addassd 11312	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + (𝐴 + 𝐶)))
10		readdridaddlidd.1	. . . . 5 ⊢ (𝜑 → (𝐵 + 𝐴) = 𝐵)
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐵 + 𝐴) = 𝐵)
12	11	oveq1d 7427	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + 𝐶))
13	9, 12	eqtr3d 2798	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶))
14	5, 7	readdcld 11319	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) ∈ ℝ)
15		readdcan 11465	. . 3 ⊢ (((𝐴 + 𝐶) ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶))
16	14, 7, 2, 15	syl3anc 1398	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶))
17	13, 16	mpbid 235	1 ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  (class class class)co 7412  ℝcr 11180   + caddc 11184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-resscn 11238  ax-addrcl 11242  ax-addass 11246  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-ltxr 11329

This theorem is used by:  reneg0addlid  43393