Theorem ltsub1 8327

Description: Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
ltsub1	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( A - C ) < ( B - C ) ) )

Proof of Theorem ltsub1

Step	Hyp	Ref	Expression
1		simp1 982	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2		simp3 984	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3		simp2 983	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
4	3, 2	resubcld 8250	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B - C ) e. RR )
5		ltsubadd 8301	. . 3 |- ( ( A e. RR /\ C e. RR /\ ( B - C ) e. RR ) -> ( ( A - C ) < ( B - C ) <-> A < ( ( B - C ) + C ) ) )
6	1, 2, 4, 5	syl3anc 1220	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - C ) < ( B - C ) <-> A < ( ( B - C ) + C ) ) )
7	3	recnd 7900	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
8	2	recnd 7900	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
9	7, 8	npcand 8184	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B - C ) + C ) = B )
10	9	breq2d 3977	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < ( ( B - C ) + C ) <-> A < B ) )
11	6, 10	bitr2d 188	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( A - C ) < ( B - C ) ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 963   e. wcel 2128   class class class wbr 3965  (class class class)co 5821   RRcr 7725   + caddc 7729   < clt 7906   - cmin 8040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-addcom 7826  ax-addass 7828  ax-distr 7830  ax-i2m1 7831  ax-0id 7834  ax-rnegex 7835  ax-cnre 7837  ax-pre-ltadd 7842

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-iota 5134  df-fun 5171  df-fv 5177  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-pnf 7908  df-mnf 7909  df-ltxr 7911  df-sub 8042  df-neg 8043

This theorem is referenced by:  lt2sub  8329  ltsub1d  8423  addltmul  9063  cos2bnd  11650