Theorem cnegexlem1 7711

Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 7714. (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 7508	. . . 4 |- ( A e. RR -> E. x e. RR ( A + x ) = 0 )
2	1	3ad2ant1 965	. . 3 |- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> E. x e. RR ( A + x ) = 0 )
3		recn 7529	. . . 4 |- ( A e. RR -> A e. CC )
4		recn 7529	. . . . . . 7 |- ( x e. RR -> x e. CC )
5		oveq2 5674	. . . . . . . . . . 11 |- ( ( A + B ) = ( A + C ) -> ( x + ( A + B ) ) = ( x + ( A + C ) ) )
6		simpr 109	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> x e. CC )
7		simpll 497	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> A e. CC )
8		simplrl 503	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> B e. CC )
9	6, 7, 8	addassd 7564	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( x + A ) + B ) = ( x + ( A + B ) ) )
10		simplrr 504	. . . . . . . . . . . . 13 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> C e. CC )
11	6, 7, 10	addassd 7564	. . . . . . . . . . . 12 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( x + A ) + C ) = ( x + ( A + C ) ) )
12	9, 11	eqeq12d 2103	. . . . . . . . . . 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	syl5ibr 155	. . . . . . . . . 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 271	. . . . . . . . 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 7673	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ x e. CC ) -> ( A + x ) = ( x + A ) )
16	15	eqeq1d 2097	. . . . . . . . . . . 12 |- ( ( A e. CC /\ x e. CC ) -> ( ( A + x ) = 0 <-> ( x + A ) = 0 ) )
17	16	adantlr 462	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) /\ x e. CC ) -> ( ( A + x ) = 0 <-> ( x + A ) = 0 ) )
18		oveq1 5673	. . . . . . . . . . . . . . 15 |- ( ( x + A ) = 0 -> ( ( x + A ) + B ) = ( 0 + B ) )
19		oveq1 5673	. . . . . . . . . . . . . . 15 |- ( ( x + A ) = 0 -> ( ( x + A ) + C ) = ( 0 + C ) )
20	18, 19	eqeq12d 2103	. . . . . . . . . . . . . 14 |- ( ( x + A ) = 0 -> ( ( ( x + A ) + B ) = ( ( x + A ) + C ) <-> ( 0 + B ) = ( 0 + C ) ) )
21	20	adantl 272	. . . . . . . . . . . . 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		addid2 7675	. . . . . . . . . . . . . . . 16 |- ( B e. CC -> ( 0 + B ) = B )
23		addid2 7675	. . . . . . . . . . . . . . . 16 |- ( C e. CC -> ( 0 + C ) = C )
24	22, 23	eqeqan12d 2104	. . . . . . . . . . . . . . 15 |- ( ( B e. CC /\ C e. CC ) -> ( ( 0 + B ) = ( 0 + C ) <-> B = C ) )
25	24	adantl 272	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) -> ( ( 0 + B ) = ( 0 + C ) <-> B = C ) )
26	25	ad2antrr 473	. . . . . . . . . . . . 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 187	. . . . . . . . . . . 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 114	. . . . . . . . . . 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 149	. . . . . . . . . 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 123	. . . . . . . . 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 148	. . . . . . . 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 114	. . . . . . 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 281	. . . . . 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 2490	. . . . 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 1140	. . . 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 1208	. . 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 5674	. 2 |- ( B = C -> ( A + B ) = ( A + C ) )
39	37, 38	impbid1 141	1 |- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 925   = wceq 1290   e. wcel 1439   E.wrex 2361  (class class class)co 5666   CCcc 7402   RRcr 7403   0cc0 7404   + caddc 7407

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-resscn 7491  ax-1cn 7492  ax-icn 7494  ax-addcl 7495  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-i2m1 7504  ax-0id 7507  ax-rnegex 7508

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036  df-ov 5669

This theorem is referenced by:  cnegexlem3  7713