MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltsubadd Structured version   Unicode version

Theorem ltsubadd 9796
Description: 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
ltsubadd  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  -  B
)  <  C  <->  A  <  ( C  +  B ) ) )

Proof of Theorem ltsubadd
StepHypRef Expression
1 simp1 981 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
2 simp2 982 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
31, 2resubcld 9763 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  -  B )  e.  RR )
4 simp3 983 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
5 ltadd1 9793 . . 3  |-  ( ( ( A  -  B
)  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  (
( A  -  B
)  <  C  <->  ( ( A  -  B )  +  B )  <  ( C  +  B )
) )
63, 4, 2, 5syl3anc 1211 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  -  B
)  <  C  <->  ( ( A  -  B )  +  B )  <  ( C  +  B )
) )
71recnd 9399 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  CC )
82recnd 9399 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  CC )
97, 8npcand 9710 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  -  B
)  +  B )  =  A )
109breq1d 4290 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( ( A  -  B )  +  B
)  <  ( C  +  B )  <->  A  <  ( C  +  B ) ) )
116, 10bitrd 253 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  -  B
)  <  C  <->  A  <  ( C  +  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 958    e. wcel 1755   class class class wbr 4280  (class class class)co 6080   RRcr 9268    + caddc 9272    < clt 9405    - cmin 9582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9407  df-mnf 9408  df-ltxr 9410  df-sub 9584  df-neg 9585
This theorem is referenced by:  ltsubadd2  9797  leaddsub  9802  ltsub23  9806  ltsubpos  9818  ltsubaddi  9884  ltsubaddd  9922  nnsub  10347  nnunb  10562  zbtwnre  10938  rebtwnz  10939  iooshf  11361  expnbnd  11976  sincosq2sgn  21845  sincosq3sgn  21846  sincosq4sgn  21847  zetacvg  26848  ftc1anclem5  28312  ftc1anclem6  28313
  Copyright terms: Public domain W3C validator