Theorem negeu 8110

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 8097	. . 3 |- ( A e. CC -> E. y e. CC ( A + y ) = 0 )
2	1	adantr 274	. 2 |- ( ( A e. CC /\ B e. CC ) -> E. y e. CC ( A + y ) = 0 )
3		simpl 108	. . . 4 |- ( ( y e. CC /\ ( A + y ) = 0 ) -> y e. CC )
4		simpr 109	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
5		addcl 7899	. . . 4 |- ( ( y e. CC /\ B e. CC ) -> ( y + B ) e. CC )
6	3, 4, 5	syl2anr 288	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) -> ( y + B ) e. CC )
7		simplrr 531	. . . . . . . 8 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> ( A + y ) = 0 )
8	7	oveq1d 5868	. . . . . . 7 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> ( ( A + y ) + B ) = ( 0 + B ) )
9		simplll 528	. . . . . . . 8 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> A e. CC )
10		simplrl 530	. . . . . . . 8 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> y e. CC )
11		simpllr 529	. . . . . . . 8 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> B e. CC )
12	9, 10, 11	addassd 7942	. . . . . . 7 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> ( ( A + y ) + B ) = ( A + ( y + B ) ) )
13	11	addid2d 8069	. . . . . . 7 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> ( 0 + B ) = B )
14	8, 12, 13	3eqtr3rd 2212	. . . . . 6 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> B = ( A + ( y + B ) ) )
15	14	eqeq2d 2182	. . . . 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 109	. . . . . 6 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) /\ x e. CC ) -> x e. CC )
17	10, 11	addcld 7939	. . . . . 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 8103	. . . . 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 187	. . . 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 2543	. . 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 2921	. . 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 409	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( A + y ) = 0 ) ) -> E! x e. CC ( A + x ) = B )
23	2, 22	rexlimddv 2592	1 |- ( ( A e. CC /\ B e. CC ) -> E! x e. CC ( A + x ) = B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   E!wreu 2450  (class class class)co 5853   CCcc 7772   0cc0 7774   + caddc 7777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  subval  8111  subcl  8118  subadd  8122