Theorem negeu 8464

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 e. CC /\ B e. CC ) -> E! x e. CC ( 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 8451	. . 3 |- ( A e. CC -> E. y e. CC ( A + y ) = 0 )
2	1	adantr 276	. 2 |- ( ( A e. CC /\ B e. CC ) -> E. y e. CC ( A + y ) = 0 )
3		simpl 109	. . . 4 |- ( ( y e. CC /\ ( A + y ) = 0 ) -> y e. CC )
4		simpr 110	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
5		addcl 8252	. . . 4 |- ( ( y e. CC /\ B e. CC ) -> ( y + B ) e. CC )
6	3, 4, 5	syl2anr 290	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) -> ( y + B ) e. CC )
7		simplrr 538	. . . . . . . 8 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> ( A + y ) = 0 )
8	7	oveq1d 6065	. . . . . . 7 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> ( ( A + y ) + B ) = ( 0 + B ) )
9		simplll 535	. . . . . . . 8 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> A e. CC )
10		simplrl 537	. . . . . . . 8 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> y e. CC )
11		simpllr 536	. . . . . . . 8 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> B e. CC )
12	9, 10, 11	addassd 8296	. . . . . . 7 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> ( ( A + y ) + B ) = ( A + ( y + B ) ) )
13	11	addlidd 8423	. . . . . . 7 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> ( 0 + B ) = B )
14	8, 12, 13	3eqtr3rd 2274	. . . . . 6 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> B = ( A + ( y + B ) ) )
15	14	eqeq2d 2244	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> ( ( A + x ) = B <-> ( A + x ) = ( A + ( y + B ) ) ) )
16		simpr 110	. . . . . 6 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> x e. CC )
17	10, 11	addcld 8293	. . . . . 6 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> ( y + B ) e. CC )
18	9, 16, 17	addcand 8457	. . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> ( ( A + x ) = ( A + ( y + B ) ) <-> x = ( y + B ) ) )
19	15, 18	bitrd 188	. . . 4 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> ( ( A + x ) = B <-> x = ( y + B ) ) )
20	19	ralrimiva 2615	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) -> A. x e. CC ( ( A + x ) = B <-> x = ( y + B ) ) )
21		reu6i 3008	. . 3 |- ( ( ( y + B ) e. CC /\ A. x e. CC ( ( A + x ) = B <-> x = ( y + B ) ) ) -> E! x e. CC ( A + x ) = B )
22	6, 20, 21	syl2anc 411	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) -> E! x e. CC ( A + x ) = B )
23	2, 22	rexlimddv 2665	1 |- ( ( A e. CC /\ B e. CC ) -> E! x e. CC ( A + x ) = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   E!wreu 2522  (class class class)co 6050   CCcc 8125   0cc0 8127   + caddc 8130

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-resscn 8219  ax-1cn 8220  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053

This theorem is referenced by:  subval  8465  subcl  8472  subadd  8476