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Theorem negeu 6959
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 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 6946 . . 3 (A ℂ → y ℂ (A + y) = 0)
21adantr 261 . 2 ((A B ℂ) → y ℂ (A + y) = 0)
3 simpl 102 . . . 4 ((y (A + y) = 0) → y ℂ)
4 simpr 103 . . . 4 ((A B ℂ) → B ℂ)
5 addcl 6764 . . . 4 ((y B ℂ) → (y + B) ℂ)
63, 4, 5syl2anr 274 . . 3 (((A B ℂ) (y (A + y) = 0)) → (y + B) ℂ)
7 simplrr 488 . . . . . . . 8 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → (A + y) = 0)
87oveq1d 5470 . . . . . . 7 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → ((A + y) + B) = (0 + B))
9 simplll 485 . . . . . . . 8 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → A ℂ)
10 simplrl 487 . . . . . . . 8 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → y ℂ)
11 simpllr 486 . . . . . . . 8 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → B ℂ)
129, 10, 11addassd 6807 . . . . . . 7 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → ((A + y) + B) = (A + (y + B)))
1311addid2d 6920 . . . . . . 7 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → (0 + B) = B)
148, 12, 133eqtr3rd 2078 . . . . . 6 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → B = (A + (y + B)))
1514eqeq2d 2048 . . . . 5 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → ((A + x) = B ↔ (A + x) = (A + (y + B))))
16 simpr 103 . . . . . 6 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → x ℂ)
1710, 11addcld 6804 . . . . . 6 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → (y + B) ℂ)
189, 16, 17addcand 6952 . . . . 5 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → ((A + x) = (A + (y + B)) ↔ x = (y + B)))
1915, 18bitrd 177 . . . 4 ((((A B ℂ) (y (A + y) = 0)) x ℂ) → ((A + x) = Bx = (y + B)))
2019ralrimiva 2386 . . 3 (((A B ℂ) (y (A + y) = 0)) → x ℂ ((A + x) = Bx = (y + B)))
21 reu6i 2726 . . 3 (((y + B) x ℂ ((A + x) = Bx = (y + B))) → ∃!x ℂ (A + x) = B)
226, 20, 21syl2anc 391 . 2 (((A B ℂ) (y (A + y) = 0)) → ∃!x ℂ (A + x) = B)
232, 22rexlimddv 2431 1 ((A B ℂ) → ∃!x ℂ (A + x) = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  wrex 2301  ∃!wreu 2302  (class class class)co 5455  cc 6669  0cc0 6671   + caddc 6674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-resscn 6735  ax-1cn 6736  ax-icn 6738  ax-addcl 6739  ax-addrcl 6740  ax-mulcl 6741  ax-addcom 6743  ax-addass 6745  ax-distr 6747  ax-i2m1 6748  ax-0id 6751  ax-rnegex 6752  ax-cnre 6754
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  subval  6960  subcl  6967  subadd  6971
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