Theorem readdcan 8159

Description: Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)

Assertion

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

Proof of Theorem readdcan

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-rnegex 7981	. . . 4 |- ( C e. RR -> E. x e. RR ( C + x ) = 0 )
2	1	3ad2ant3 1022	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. x e. RR ( C + x ) = 0 )
3		oveq2 5926	. . . . . . 7 |- ( ( C + A ) = ( C + B ) -> ( x + ( C + A ) ) = ( x + ( C + B ) ) )
4	3	adantl 277	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( x + ( C + A ) ) = ( x + ( C + B ) ) )
5		simprl 529	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> x e. RR )
6	5	recnd 8048	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> x e. CC )
7		simpl3 1004	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. RR )
8	7	recnd 8048	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. CC )
9		simpl1 1002	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. RR )
10	9	recnd 8048	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. CC )
11	6, 8, 10	addassd 8042	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + A ) = ( x + ( C + A ) ) )
12		simpl2 1003	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. RR )
13	12	recnd 8048	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. CC )
14	6, 8, 13	addassd 8042	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + B ) = ( x + ( C + B ) ) )
15	11, 14	eqeq12d 2208	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( ( x + C ) + A ) = ( ( x + C ) + B ) <-> ( x + ( C + A ) ) = ( x + ( C + B ) ) ) )
16	15	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( ( ( x + C ) + A ) = ( ( x + C ) + B ) <-> ( x + ( C + A ) ) = ( x + ( C + B ) ) ) )
17	4, 16	mpbird 167	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( ( x + C ) + A ) = ( ( x + C ) + B ) )
18	8	adantr 276	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> C e. CC )
19	6	adantr 276	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> x e. CC )
20		addcom 8156	. . . . . . . . 9 |- ( ( C e. CC /\ x e. CC ) -> ( C + x ) = ( x + C ) )
21	18, 19, 20	syl2anc 411	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( C + x ) = ( x + C ) )
22		simplrr 536	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( C + x ) = 0 )
23	21, 22	eqtr3d 2228	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( x + C ) = 0 )
24	23	oveq1d 5933	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( ( x + C ) + A ) = ( 0 + A ) )
25	10	adantr 276	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> A e. CC )
26		addlid 8158	. . . . . . 7 |- ( A e. CC -> ( 0 + A ) = A )
27	25, 26	syl 14	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( 0 + A ) = A )
28	24, 27	eqtrd 2226	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( ( x + C ) + A ) = A )
29	23	oveq1d 5933	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( ( x + C ) + B ) = ( 0 + B ) )
30	13	adantr 276	. . . . . . 7 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> B e. CC )
31		addlid 8158	. . . . . . 7 |- ( B e. CC -> ( 0 + B ) = B )
32	30, 31	syl 14	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( 0 + B ) = B )
33	29, 32	eqtrd 2226	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> ( ( x + C ) + B ) = B )
34	17, 28, 33	3eqtr3d 2234	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) /\ ( C + A ) = ( C + B ) ) -> A = B )
35	34	ex 115	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( C + A ) = ( C + B ) -> A = B ) )
36	2, 35	rexlimddv 2616	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) = ( C + B ) -> A = B ) )
37		oveq2 5926	. 2 |- ( A = B -> ( C + A ) = ( C + B ) )
38	36, 37	impbid1 142	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) = ( C + B ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473  (class class class)co 5918   CCcc 7870   RRcr 7871   0cc0 7872   + caddc 7875

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-resscn 7964  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by: (None)