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Theorem negeu 7824
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 7811 . . 3 (𝐴 ∈ ℂ → ∃𝑦 ∈ ℂ (𝐴 + 𝑦) = 0)
21adantr 272 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃𝑦 ∈ ℂ (𝐴 + 𝑦) = 0)
3 simpl 108 . . . 4 ((𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0) → 𝑦 ∈ ℂ)
4 simpr 109 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
5 addcl 7617 . . . 4 ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑦 + 𝐵) ∈ ℂ)
63, 4, 5syl2anr 286 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → (𝑦 + 𝐵) ∈ ℂ)
7 simplrr 506 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → (𝐴 + 𝑦) = 0)
87oveq1d 5721 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑦) + 𝐵) = (0 + 𝐵))
9 simplll 503 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
10 simplrl 505 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝑦 ∈ ℂ)
11 simpllr 504 . . . . . . . 8 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)
129, 10, 11addassd 7660 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑦) + 𝐵) = (𝐴 + (𝑦 + 𝐵)))
1311addid2d 7783 . . . . . . 7 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → (0 + 𝐵) = 𝐵)
148, 12, 133eqtr3rd 2141 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝐵 = (𝐴 + (𝑦 + 𝐵)))
1514eqeq2d 2111 . . . . 5 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 𝐵 ↔ (𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝐵))))
16 simpr 109 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
1710, 11addcld 7657 . . . . . 6 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → (𝑦 + 𝐵) ∈ ℂ)
189, 16, 17addcand 7817 . . . . 5 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝐵)) ↔ 𝑥 = (𝑦 + 𝐵)))
1915, 18bitrd 187 . . . 4 ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 𝐵𝑥 = (𝑦 + 𝐵)))
2019ralrimiva 2464 . . 3 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → ∀𝑥 ∈ ℂ ((𝐴 + 𝑥) = 𝐵𝑥 = (𝑦 + 𝐵)))
21 reu6i 2828 . . 3 (((𝑦 + 𝐵) ∈ ℂ ∧ ∀𝑥 ∈ ℂ ((𝐴 + 𝑥) = 𝐵𝑥 = (𝑦 + 𝐵))) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
226, 20, 21syl2anc 406 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
232, 22rexlimddv 2513 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1299  wcel 1448  wral 2375  wrex 2376  ∃!wreu 2377  (class class class)co 5706  cc 7498  0cc0 7500   + caddc 7503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-resscn 7587  ax-1cn 7588  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-reu 2382  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-iota 5024  df-fv 5067  df-ov 5709
This theorem is referenced by:  subval  7825  subcl  7832  subadd  7836
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