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Theorem readdridaddlidd 42299
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 11435, 𝐴 is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.)
Hypotheses
Ref Expression
readdridaddlidd.a (𝜑𝐴 ∈ ℝ)
readdridaddlidd.b (𝜑𝐵 ∈ ℝ)
readdridaddlidd.1 (𝜑 → (𝐵 + 𝐴) = 𝐵)
Assertion
Ref Expression
readdridaddlidd ((𝜑𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶)

Proof of Theorem readdridaddlidd
StepHypRef Expression
1 readdridaddlidd.b . . . . . 6 (𝜑𝐵 ∈ ℝ)
21adantr 480 . . . . 5 ((𝜑𝐶 ∈ ℝ) → 𝐵 ∈ ℝ)
32recnd 11289 . . . 4 ((𝜑𝐶 ∈ ℝ) → 𝐵 ∈ ℂ)
4 readdridaddlidd.a . . . . . 6 (𝜑𝐴 ∈ ℝ)
54adantr 480 . . . . 5 ((𝜑𝐶 ∈ ℝ) → 𝐴 ∈ ℝ)
65recnd 11289 . . . 4 ((𝜑𝐶 ∈ ℝ) → 𝐴 ∈ ℂ)
7 simpr 484 . . . . 5 ((𝜑𝐶 ∈ ℝ) → 𝐶 ∈ ℝ)
87recnd 11289 . . . 4 ((𝜑𝐶 ∈ ℝ) → 𝐶 ∈ ℂ)
93, 6, 8addassd 11283 . . 3 ((𝜑𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + (𝐴 + 𝐶)))
10 readdridaddlidd.1 . . . . 5 (𝜑 → (𝐵 + 𝐴) = 𝐵)
1110adantr 480 . . . 4 ((𝜑𝐶 ∈ ℝ) → (𝐵 + 𝐴) = 𝐵)
1211oveq1d 7446 . . 3 ((𝜑𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + 𝐶))
139, 12eqtr3d 2779 . 2 ((𝜑𝐶 ∈ ℝ) → (𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶))
145, 7readdcld 11290 . . 3 ((𝜑𝐶 ∈ ℝ) → (𝐴 + 𝐶) ∈ ℝ)
15 readdcan 11435 . . 3 (((𝐴 + 𝐶) ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶))
1614, 7, 2, 15syl3anc 1373 . 2 ((𝜑𝐶 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶))
1713, 16mpbid 232 1 ((𝜑𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  (class class class)co 7431  cr 11154   + caddc 11158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-addrcl 11216  ax-addass 11220  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-ltxr 11300
This theorem is referenced by:  reneg0addlid  42404
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