Theorem readdcan 8087

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 7911	. . . 4 |- ( C e. RR -> E. x e. RR ( C + x ) = 0 )
2	1	3ad2ant3 1020	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. x e. RR ( C + x ) = 0 )
3		oveq2 5877	. . . . . . 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 7976	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> x e. CC )
7		simpl3 1002	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. RR )
8	7	recnd 7976	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. CC )
9		simpl1 1000	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. RR )
10	9	recnd 7976	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. CC )
11	6, 8, 10	addassd 7970	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + A ) = ( x + ( C + A ) ) )
12		simpl2 1001	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. RR )
13	12	recnd 7976	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. CC )
14	6, 8, 13	addassd 7970	. . . . . . . 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 2192	. . . . . . 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 8084	. . . . . . . . 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 2212	. . . . . . 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 5884	. . . . . 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		addid2 8086	. . . . . . 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 2210	. . . . 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 5884	. . . . . 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		addid2 8086	. . . . . . 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 2210	. . . . 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 2218	. . . 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 2599	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) = ( C + B ) -> A = B ) )
37		oveq2 5877	. 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 978   = wceq 1353   e. wcel 2148   E.wrex 2456  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   + caddc 7805

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-resscn 7894  ax-1cn 7895  ax-icn 7897  ax-addcl 7898  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-i2m1 7907  ax-0id 7910  ax-rnegex 7911

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220  df-ov 5872

This theorem is referenced by: (None)