Theorem negeu 8162

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.

Step	Hyp	Ref	Expression
1		cnegex 8149	. . 3 ⊢ (𝐴 ∈ ℂ → ∃𝑦 ∈ ℂ (𝐴 + 𝑦) = 0)
2	1	adantr 276	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃𝑦 ∈ ℂ (𝐴 + 𝑦) = 0)
3		simpl 109	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0) → 𝑦 ∈ ℂ)
4		simpr 110	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
5		addcl 7950	. . . 4 ⊢ ((𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑦 + 𝐵) ∈ ℂ)
6	3, 4, 5	syl2anr 290	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → (𝑦 + 𝐵) ∈ ℂ)
7		simplrr 536	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → (𝐴 + 𝑦) = 0)
8	7	oveq1d 5903	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑦) + 𝐵) = (0 + 𝐵))
9		simplll 533	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
10		simplrl 535	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝑦 ∈ ℂ)
11		simpllr 534	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)
12	9, 10, 11	addassd 7994	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑦) + 𝐵) = (𝐴 + (𝑦 + 𝐵)))
13	11	addid2d 8121	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → (0 + 𝐵) = 𝐵)
14	8, 12, 13	3eqtr3rd 2229	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝐵 = (𝐴 + (𝑦 + 𝐵)))
15	14	eqeq2d 2199	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 𝐵 ↔ (𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝐵))))
16		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
17	10, 11	addcld 7991	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → (𝑦 + 𝐵) ∈ ℂ)
18	9, 16, 17	addcand 8155	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = (𝐴 + (𝑦 + 𝐵)) ↔ 𝑥 = (𝑦 + 𝐵)))
19	15, 18	bitrd 188	. . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) ∧ 𝑥 ∈ ℂ) → ((𝐴 + 𝑥) = 𝐵 ↔ 𝑥 = (𝑦 + 𝐵)))
20	19	ralrimiva 2560	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → ∀𝑥 ∈ ℂ ((𝐴 + 𝑥) = 𝐵 ↔ 𝑥 = (𝑦 + 𝐵)))
21		reu6i 2940	. . 3 ⊢ (((𝑦 + 𝐵) ∈ ℂ ∧ ∀𝑥 ∈ ℂ ((𝐴 + 𝑥) = 𝐵 ↔ 𝑥 = (𝑦 + 𝐵))) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
22	6, 20, 21	syl2anc 411	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝐴 + 𝑦) = 0)) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
23	2, 22	rexlimddv 2609	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1363   ∈ wcel 2158  ∀wral 2465  ∃wrex 2466  ∃!wreu 2467  (class class class)co 5888  ℂcc 7823  0cc0 7825   + caddc 7828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-resscn 7917  ax-1cn 7918  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-distr 7929  ax-i2m1 7930  ax-0id 7933  ax-rnegex 7934  ax-cnre 7936

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891

This theorem is referenced by:  subval  8163  subcl  8170  subadd  8174