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Theorem readdridaddlidd 42278
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 11433, 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 11287 . . . 4 ((𝜑𝐶 ∈ ℝ) → 𝐵 ∈ ℂ)
4 readdridaddlidd.a . . . . . 6 (𝜑𝐴 ∈ ℝ)
54adantr 480 . . . . 5 ((𝜑𝐶 ∈ ℝ) → 𝐴 ∈ ℝ)
65recnd 11287 . . . 4 ((𝜑𝐶 ∈ ℝ) → 𝐴 ∈ ℂ)
7 simpr 484 . . . . 5 ((𝜑𝐶 ∈ ℝ) → 𝐶 ∈ ℝ)
87recnd 11287 . . . 4 ((𝜑𝐶 ∈ ℝ) → 𝐶 ∈ ℂ)
93, 6, 8addassd 11281 . . 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 2777 . 2 ((𝜑𝐶 ∈ ℝ) → (𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶))
145, 7readdcld 11288 . . 3 ((𝜑𝐶 ∈ ℝ) → (𝐴 + 𝐶) ∈ ℝ)
15 readdcan 11433 . . 3 (((𝐴 + 𝐶) ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶))
1614, 7, 2, 15syl3anc 1370 . 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 1537  wcel 2106  (class class class)co 7431  cr 11152   + caddc 11156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210  ax-addrcl 11214  ax-addass 11218  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-ltxr 11298
This theorem is referenced by:  reneg0addlid  42381
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