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Theorem ltadd2 9133
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 9105 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) ) )
2 oveq2 6048 . . . . . 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 9105 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( C  +  B )  <  ( C  +  A
) ) )
543com12 1157 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( C  +  B )  <  ( C  +  A
) ) )
63, 5orim12d 812 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  =  B  \/  B  <  A
)  ->  ( ( C  +  A )  =  ( C  +  B )  \/  ( C  +  B )  <  ( C  +  A
) ) ) )
76con3d 127 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -.  ( ( C  +  A )  =  ( C  +  B )  \/  ( C  +  B )  <  ( C  +  A )
)  ->  -.  ( A  =  B  \/  B  <  A ) ) )
8 simp3 959 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
9 simp1 957 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
108, 9readdcld 9071 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  +  A )  e.  RR )
11 simp2 958 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
128, 11readdcld 9071 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  +  B )  e.  RR )
13 axlttri 9103 . . . 4  |-  ( ( ( C  +  A
)  e.  RR  /\  ( C  +  B
)  e.  RR )  ->  ( ( C  +  A )  < 
( C  +  B
)  <->  -.  ( ( C  +  A )  =  ( C  +  B )  \/  ( C  +  B )  <  ( C  +  A
) ) ) )
1410, 12, 13syl2anc 643 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  <->  -.  (
( C  +  A
)  =  ( C  +  B )  \/  ( C  +  B
)  <  ( C  +  A ) ) ) )
15 axlttri 9103 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A
) ) )
169, 11, 15syl2anc 643 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )
177, 14, 163imtr4d 260 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  ->  A  <  B ) )
181, 17impbid 184 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172  (class class class)co 6040   RRcr 8945    + caddc 8949    < clt 9076
This theorem is referenced by:  ltadd2d  9182  readdcan  9196  ltadd1  9451  ltaddpos  9474  avglt1  10161  flbi2  11179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-addrcl 9007  ax-pre-lttri 9020  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081
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