MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltadd2 Structured version   Unicode version

Theorem ltadd2 9720
Description: Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
ltadd2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B )
) )

Proof of Theorem ltadd2
StepHypRef Expression
1 axltadd 9689 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) ) )
2 oveq2 6286 . . . . . 6  |-  ( A  =  B  ->  ( C  +  A )  =  ( C  +  B ) )
32a1i 11 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  =  B  ->  ( C  +  A )  =  ( C  +  B ) ) )
4 axltadd 9689 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( C  +  B )  <  ( C  +  A
) ) )
543com12 1201 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( C  +  B )  <  ( C  +  A
) ) )
63, 5orim12d 839 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  =  B  \/  B  <  A
)  ->  ( ( C  +  A )  =  ( C  +  B )  \/  ( C  +  B )  <  ( C  +  A
) ) ) )
76con3d 133 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -.  ( ( C  +  A )  =  ( C  +  B )  \/  ( C  +  B )  <  ( C  +  A )
)  ->  -.  ( A  =  B  \/  B  <  A ) ) )
8 simp3 999 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
9 simp1 997 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
108, 9readdcld 9653 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  +  A )  e.  RR )
11 simp2 998 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
128, 11readdcld 9653 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  +  B )  e.  RR )
13 axlttri 9687 . . . 4  |-  ( ( ( C  +  A
)  e.  RR  /\  ( C  +  B
)  e.  RR )  ->  ( ( C  +  A )  < 
( C  +  B
)  <->  -.  ( ( C  +  A )  =  ( C  +  B )  \/  ( C  +  B )  <  ( C  +  A
) ) ) )
1410, 12, 13syl2anc 659 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  <->  -.  (
( C  +  A
)  =  ( C  +  B )  \/  ( C  +  B
)  <  ( C  +  A ) ) ) )
15 axlttri 9687 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A
) ) )
169, 11, 15syl2anc 659 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )
177, 14, 163imtr4d 268 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  ->  A  <  B ) )
181, 17impbid 190 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4395  (class class class)co 6278   RRcr 9521    + caddc 9525    < clt 9658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-addrcl 9583  ax-pre-lttri 9596  ax-pre-ltadd 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-ltxr 9663
This theorem is referenced by:  ltadd2i  9747  ltadd2d  9772  readdcan  9788  ltadd1  10060  ltaddpos  10083  ltaddsublt  10217  avglt1  10817  flbi2  11990  dvtanlemOLD  31437  ltaddneg  36856
  Copyright terms: Public domain W3C validator