Mathbox for Steven Nguyen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  readdid1addid2d Structured version   Visualization version   GIF version

 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 10810, 𝐴 is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.)
Hypotheses
Ref Expression
Assertion
Ref Expression

StepHypRef Expression
21adantr 484 . . . . 5 ((𝜑𝐶 ∈ ℝ) → 𝐵 ∈ ℝ)
32recnd 10665 . . . 4 ((𝜑𝐶 ∈ ℝ) → 𝐵 ∈ ℂ)
54adantr 484 . . . . 5 ((𝜑𝐶 ∈ ℝ) → 𝐴 ∈ ℝ)
65recnd 10665 . . . 4 ((𝜑𝐶 ∈ ℝ) → 𝐴 ∈ ℂ)
7 simpr 488 . . . . 5 ((𝜑𝐶 ∈ ℝ) → 𝐶 ∈ ℝ)
87recnd 10665 . . . 4 ((𝜑𝐶 ∈ ℝ) → 𝐶 ∈ ℂ)
93, 6, 8addassd 10659 . . 3 ((𝜑𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + (𝐴 + 𝐶)))
10 readdid1addid2d.1 . . . . 5 (𝜑 → (𝐵 + 𝐴) = 𝐵)
1110adantr 484 . . . 4 ((𝜑𝐶 ∈ ℝ) → (𝐵 + 𝐴) = 𝐵)
1211oveq1d 7155 . . 3 ((𝜑𝐶 ∈ ℝ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + 𝐶))
139, 12eqtr3d 2835 . 2 ((𝜑𝐶 ∈ ℝ) → (𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶))
145, 7readdcld 10666 . . 3 ((𝜑𝐶 ∈ ℝ) → (𝐴 + 𝐶) ∈ ℝ)
15 readdcan 10810 . . 3 (((𝐴 + 𝐶) ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶))
1614, 7, 2, 15syl3anc 1368 . 2 ((𝜑𝐶 ∈ ℝ) → ((𝐵 + (𝐴 + 𝐶)) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐶) = 𝐶))
1713, 16mpbid 235 1 ((𝜑𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  (class class class)co 7140  ℝcr 10532   + caddc 10536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448  ax-resscn 10590  ax-addrcl 10594  ax-addass 10598  ax-pre-lttri 10607  ax-pre-lttrn 10608  ax-pre-ltadd 10609 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-po 5439  df-so 5440  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7143  df-er 8279  df-en 8500  df-dom 8501  df-sdom 8502  df-pnf 10673  df-mnf 10674  df-ltxr 10676 This theorem is referenced by:  reneg0addid2  39576
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