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Theorem ltadd2i 9160
Description: Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Paul Chapman, 27-Jan-2008.)
Hypotheses
Ref Expression
lt.1  |-  A  e.  RR
lt.2  |-  B  e.  RR
lt.3  |-  C  e.  RR
Assertion
Ref Expression
ltadd2i  |-  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B )
)

Proof of Theorem ltadd2i
StepHypRef Expression
1 lt.1 . . 3  |-  A  e.  RR
2 lt.2 . . 3  |-  B  e.  RR
3 lt.3 . . 3  |-  C  e.  RR
4 axltadd 9105 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) ) )
51, 2, 3, 4mp3an 1279 . 2  |-  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) )
6 axltadd 9105 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( C  +  B )  <  ( C  +  A
) ) )
72, 1, 3, 6mp3an 1279 . . . . . 6  |-  ( B  <  A  ->  ( C  +  B )  <  ( C  +  A
) )
8 oveq2 6048 . . . . . 6  |-  ( B  =  A  ->  ( C  +  B )  =  ( C  +  A ) )
97, 8orim12i 503 . . . . 5  |-  ( ( B  <  A  \/  B  =  A )  ->  ( ( C  +  B )  <  ( C  +  A )  \/  ( C  +  B
)  =  ( C  +  A ) ) )
102, 1leloei 9146 . . . . 5  |-  ( B  <_  A  <->  ( B  <  A  \/  B  =  A ) )
113, 2readdcli 9059 . . . . . 6  |-  ( C  +  B )  e.  RR
123, 1readdcli 9059 . . . . . 6  |-  ( C  +  A )  e.  RR
1311, 12leloei 9146 . . . . 5  |-  ( ( C  +  B )  <_  ( C  +  A )  <->  ( ( C  +  B )  <  ( C  +  A
)  \/  ( C  +  B )  =  ( C  +  A
) ) )
149, 10, 133imtr4i 258 . . . 4  |-  ( B  <_  A  ->  ( C  +  B )  <_  ( C  +  A
) )
152, 1lenlti 9149 . . . 4  |-  ( B  <_  A  <->  -.  A  <  B )
1611, 12lenlti 9149 . . . 4  |-  ( ( C  +  B )  <_  ( C  +  A )  <->  -.  ( C  +  A )  <  ( C  +  B
) )
1714, 15, 163imtr3i 257 . . 3  |-  ( -.  A  <  B  ->  -.  ( C  +  A
)  <  ( C  +  B ) )
1817con4i 124 . 2  |-  ( ( C  +  A )  <  ( C  +  B )  ->  A  <  B )
195, 18impbii 181 1  |-  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    = wceq 1649    e. wcel 1721   class class class wbr 4172  (class class class)co 6040   RRcr 8945    + caddc 8949    < clt 9076    <_ cle 9077
This theorem is referenced by:  numlt  10357  bposlem8  21028
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-xr 9080  df-ltxr 9081  df-le 9082
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