Theorem ltsub1 8429

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 998	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2		simp3 1000	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3		simp2 999	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
4	3, 2	resubcld 8352	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B - C ) e. RR )
5		ltsubadd 8403	. . 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 1248	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - C ) < ( B - C ) <-> A < ( ( B - C ) + C ) ) )
7	3	recnd 8000	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
8	2	recnd 8000	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
9	7, 8	npcand 8286	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B - C ) + C ) = B )
10	9	breq2d 4027	. 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 979   e. wcel 2158   class class class wbr 4015  (class class class)co 5888   RRcr 7824   + caddc 7828   < clt 8006   - cmin 8142

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-distr 7929  ax-i2m1 7930  ax-0id 7933  ax-rnegex 7934  ax-cnre 7936  ax-pre-ltadd 7941

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-sub 8144  df-neg 8145

This theorem is referenced by:  lt2sub  8431  ltsub1d  8525  addltmul  9169  cos2bnd  11782