Theorem cnegexlem1 8282

Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 8285. (Contributed by Eric Schmidt, 22-May-2007.)

Assertion

Ref	Expression
cnegexlem1	|- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )

Proof of Theorem cnegexlem1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-rnegex 8069	. . . 4 |- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
2	1	3ad2ant1 1021	. . 3 |- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> E. x e. RR ( A + x ) = 0 )
3		recn 8093	. . . 4 |- ( A e. RR -> A e. CC )
4		recn 8093	. . . . . . 7 |- ( x e. RR -> x e. CC )
5		oveq2 5975	. . . . . . . . . . 11 |- ( ( A + B ) = ( A + C ) -> ( x + ( A + B ) ) = ( x + ( A + C ) ) )
6		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> x e. CC )
7		simpll 527	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> A e. CC )
8		simplrl 535	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> B e. CC )
9	6, 7, 8	addassd 8130	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( x + A ) + B ) = ( x + ( A + B ) ) )
10		simplrr 536	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> C e. CC )
11	6, 7, 10	addassd 8130	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( x + A ) + C ) = ( x + ( A + C ) ) )
12	9, 11	eqeq12d 2222	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> ( x + ( A + B ) ) = ( x + ( A + C ) ) ) )
13	5, 12	imbitrrid 156	. . . . . . . . . 10 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( A + B ) = ( A + C ) -> ( ( x + A ) + B ) = ( ( x + A ) + C ) ) )
14	13	adantr 276	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) /\ ( A + x ) = 0 ) -> ( ( A + B ) = ( A + C ) -> ( ( x + A ) + B ) = ( ( x + A ) + C ) ) )
15		addcom 8244	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ x e. CC ) -> ( A + x ) = ( x + A ) )
16	15	eqeq1d 2216	. . . . . . . . . . . 12 |- ( ( A e. CC /\ x e. CC ) -> ( ( A + x ) = 0 <-> ( x + A ) = 0 ) )
17	16	adantlr 477	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( A + x ) = 0 <-> ( x + A ) = 0 ) )
18		oveq1 5974	. . . . . . . . . . . . . . 15 |- ( ( x + A ) = 0 -> ( ( x + A ) + B ) = ( 0 + B ) )
19		oveq1 5974	. . . . . . . . . . . . . . 15 |- ( ( x + A ) = 0 -> ( ( x + A ) + C ) = ( 0 + C ) )
20	18, 19	eqeq12d 2222	. . . . . . . . . . . . . 14 |- ( ( x + A ) = 0 -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> ( 0 + B ) = ( 0 + C ) ) )
21	20	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) /\ ( x + A ) = 0 ) -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> ( 0 + B ) = ( 0 + C ) ) )
22		addlid 8246	. . . . . . . . . . . . . . . 16 |- ( B e. CC -> ( 0 + B ) = B )
23		addlid 8246	. . . . . . . . . . . . . . . 16 |- ( C e. CC -> ( 0 + C ) = C )
24	22, 23	eqeqan12d 2223	. . . . . . . . . . . . . . 15 |- ( ( B e. CC /\ C e. CC ) -> ( ( 0 + B ) = ( 0 + C ) <-> B = C ) )
25	24	adantl 277	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) -> ( ( 0 + B ) = ( 0 + C ) <-> B = C ) )
26	25	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) /\ ( x + A ) = 0 ) -> ( ( 0 + B ) = ( 0 + C ) <-> B = C ) )
27	21, 26	bitrd 188	. . . . . . . . . . . 12 |- ( ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) /\ ( x + A ) = 0 ) -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> B = C ) )
28	27	ex 115	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( x + A ) = 0 -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> B = C ) ) )
29	17, 28	sylbid 150	. . . . . . . . . 10 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( A + x ) = 0 -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> B = C ) ) )
30	29	imp 124	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) /\ ( A + x ) = 0 ) -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> B = C ) )
31	14, 30	sylibd 149	. . . . . . . 8 |- ( ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) /\ ( A + x ) = 0 ) -> ( ( A + B ) = ( A + C ) -> B = C ) )
32	31	ex 115	. . . . . . 7 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( A + x ) = 0 -> ( ( A + B ) = ( A + C ) -> B = C ) ) )
33	4, 32	sylan2 286	. . . . . 6 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. RR ) -> ( ( A + x ) = 0 -> ( ( A + B ) = ( A + C ) -> B = C ) ) )
34	33	rexlimdva 2625	. . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) -> ( E. x e. RR ( A + x ) = 0 -> ( ( A + B ) = ( A + C ) -> B = C ) ) )
35	34	3impb 1202	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( E. x e. RR ( A + x ) = 0 -> ( ( A + B ) = ( A + C ) -> B = C ) ) )
36	3, 35	syl3an1 1283	. . 3 |- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( E. x e. RR ( A + x ) = 0 -> ( ( A + B ) = ( A + C ) -> B = C ) ) )
37	2, 36	mpd 13	. 2 |- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) -> B = C ) )
38		oveq2 5975	. 2 |- ( B = C -> ( A + B ) = ( A + C ) )
39	37, 38	impbid1 142	1 |- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   E.wrex 2487  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   + caddc 7963

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-resscn 8052  ax-1cn 8053  ax-icn 8055  ax-addcl 8056  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by:  cnegexlem3  8284