Theorem ltsub1 8415

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 997	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2		simp3 999	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3		simp2 998	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
4	3, 2	resubcld 8338	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B - C ) e. RR )
5		ltsubadd 8389	. . 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 1238	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - C ) < ( B - C ) <-> A < ( ( B - C ) + C ) ) )
7	3	recnd 7986	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
8	2	recnd 7986	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
9	7, 8	npcand 8272	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B - C ) + C ) = B )
10	9	breq2d 4016	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < ( ( B - C ) + C ) <-> A < B ) )
11	6, 10	bitr2d 189	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( A - C ) < ( B - C ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 978   e. wcel 2148   class class class wbr 4004  (class class class)co 5875   RRcr 7810   + caddc 7814   < clt 7992   - cmin 8128

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-in1 614  ax-in2 615  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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-sub 8130  df-neg 8131

This theorem is referenced by:  lt2sub  8417  ltsub1d  8511  addltmul  9155  cos2bnd  11768