Theorem readdcan 9271

Description: Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

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

Proof of Theorem readdcan

Step	Hyp	Ref	Expression
1		ltadd2 9208	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C + A ) < ( C + B ) ) )
2	1	notbid 287	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( -. A < B <-> -. ( C + A ) < ( C + B ) ) )
3		simp2 959	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
4		simp1 958	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
5		simp3 960	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
6	3, 4, 5	ltadd2d 9257	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B < A <-> ( C + B ) < ( C + A ) ) )
7	6	notbid 287	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( -. B < A <-> -. ( C + B ) < ( C + A ) ) )
8	2, 7	anbi12d 693	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( -. A < B /\ -. B < A ) <-> ( -. ( C + A ) < ( C + B ) /\ -. ( C + B ) < ( C + A ) ) ) )
9	4, 3	lttri3d 9244	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
10	5, 4	readdcld 9146	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C + A ) e. RR )
11	5, 3	readdcld 9146	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C + B ) e. RR )
12	10, 11	lttri3d 9244	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) = ( C + B ) <-> ( -. ( C + A ) < ( C + B ) /\ -. ( C + B ) < ( C + A ) ) ) )
13	8, 9, 12	3bitr4rd 279	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) = ( C + B ) <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 178   /\ wa 360   /\ w3a 937   = wceq 1653   e. wcel 1727   class class class wbr 4237  (class class class)co 6110   RRcr 9020   + caddc 9024   < clt 9151

This theorem is referenced by:  00id  9272  mul02lem2  9274  addid1  9277  rpnnen2lem10  12854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-resscn 9078  ax-addrcl 9082  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-ltxr 9156