ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  negeu GIF version

Theorem negeu 7264
Description: Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
negeu ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem negeu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnegex 7251 . . 3 (𝐴 ∈ ℂ → ∃𝑦 ∈ ℂ (𝐴 + 𝑦) = 0)
21adantr 265 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃𝑦 ∈ ℂ (𝐴 + 𝑦) = 0)
3 simpl 106 . . . 4 ((𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0) → 𝑦 ∈ ℂ)
4 simpr 107 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
5 addcl 7063 . . . 4 ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑦 + 𝐵) ∈ ℂ)
63, 4, 5syl2anr 278 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → (𝑦 + 𝐵) ∈ ℂ)
7 simplrr 496 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → (𝐴 + 𝑦) = 0)
87oveq1d 5554 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑦) + 𝐵) = (0 + 𝐵))
9 simplll 493 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
10 simplrl 495 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝑦 ∈ ℂ)
11 simpllr 494 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)
129, 10, 11addassd 7106 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑦) + 𝐵) = (𝐴 + (𝑦 + 𝐵)))
1311addid2d 7223 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → (0 + 𝐵) = 𝐵)
148, 12, 133eqtr3rd 2097 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝐵 = (𝐴 + (𝑦 + 𝐵)))
1514eqeq2d 2067 . . . . 5 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 𝐵 ↔ (𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝐵))))
16 simpr 107 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
1710, 11addcld 7103 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → (𝑦 + 𝐵) ∈ ℂ)
189, 16, 17addcand 7257 . . . . 5 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝐵)) ↔ 𝑥 = (𝑦 + 𝐵)))
1915, 18bitrd 181 . . . 4 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 𝐵𝑥 = (𝑦 + 𝐵)))
2019ralrimiva 2409 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → ∀𝑥 ∈ ℂ ((𝐴 + 𝑥) = 𝐵𝑥 = (𝑦 + 𝐵)))
21 reu6i 2754 . . 3 (((𝑦 + 𝐵) ∈ ℂ ∧ ∀𝑥 ∈ ℂ ((𝐴 + 𝑥) = 𝐵𝑥 = (𝑦 + 𝐵))) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
226, 20, 21syl2anc 397 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
232, 22rexlimddv 2454 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323  wrex 2324  ∃!wreu 2325  (class class class)co 5539  cc 6944  0cc0 6946   + caddc 6949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-resscn 7033  ax-1cn 7034  ax-icn 7036  ax-addcl 7037  ax-addrcl 7038  ax-mulcl 7039  ax-addcom 7041  ax-addass 7043  ax-distr 7045  ax-i2m1 7046  ax-0id 7049  ax-rnegex 7050  ax-cnre 7052
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-iota 4894  df-fv 4937  df-ov 5542
This theorem is referenced by:  subval  7265  subcl  7272  subadd  7276
  Copyright terms: Public domain W3C validator