Theorem readdcan 9204

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 9141	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C + A ) < ( C + B ) ) )
2	1	notbid 286	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( -. A < B <-> -. ( C + A ) < ( C + B ) ) )
3		simp2 958	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
4		simp1 957	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
5		simp3 959	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
6	3, 4, 5	ltadd2d 9190	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B < A <-> ( C + B ) < ( C + A ) ) )
7	6	notbid 286	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( -. B < A <-> -. ( C + B ) < ( C + A ) ) )
8	2, 7	anbi12d 692	. 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 9177	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
10	5, 4	readdcld 9079	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C + A ) e. RR )
11	5, 3	readdcld 9079	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C + B ) e. RR )
12	10, 11	lttri3d 9177	. 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 278	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) = ( C + B ) <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   class class class wbr 4180  (class class class)co 6048   RRcr 8953   + caddc 8957   < clt 9084

This theorem is referenced by:  00id  9205  mul02lem2  9207  addid1  9210  rpnnen2lem10  12786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-addrcl 9015  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-ltxr 9089