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Theorem ltadd2i 8950
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 8896 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) ) )
51, 2, 3, 4mp3an 1277 . 2  |-  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) )
6 axltadd 8896 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( C  +  B )  <  ( C  +  A
) ) )
72, 1, 3, 6mp3an 1277 . . . . . 6  |-  ( B  <  A  ->  ( C  +  B )  <  ( C  +  A
) )
8 oveq2 5866 . . . . . 6  |-  ( B  =  A  ->  ( C  +  B )  =  ( C  +  A ) )
97, 8orim12i 502 . . . . 5  |-  ( ( B  <  A  \/  B  =  A )  ->  ( ( C  +  B )  <  ( C  +  A )  \/  ( C  +  B
)  =  ( C  +  A ) ) )
102, 1leloei 8935 . . . . 5  |-  ( B  <_  A  <->  ( B  <  A  \/  B  =  A ) )
113, 2readdcli 8850 . . . . . 6  |-  ( C  +  B )  e.  RR
123, 1readdcli 8850 . . . . . 6  |-  ( C  +  A )  e.  RR
1311, 12leloei 8935 . . . . 5  |-  ( ( C  +  B )  <_  ( C  +  A )  <->  ( ( C  +  B )  <  ( C  +  A
)  \/  ( C  +  B )  =  ( C  +  A
) ) )
149, 10, 133imtr4i 257 . . . 4  |-  ( B  <_  A  ->  ( C  +  B )  <_  ( C  +  A
) )
152, 1lenlti 8938 . . . 4  |-  ( B  <_  A  <->  -.  A  <  B )
1611, 12lenlti 8938 . . . 4  |-  ( ( C  +  B )  <_  ( C  +  A )  <->  -.  ( C  +  A )  <  ( C  +  B
) )
1714, 15, 163imtr3i 256 . . 3  |-  ( -.  A  <  B  ->  -.  ( C  +  A
)  <  ( C  +  B ) )
1817con4i 122 . 2  |-  ( ( C  +  A )  <  ( C  +  B )  ->  A  <  B )
195, 18impbii 180 1  |-  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   RRcr 8736    + caddc 8740    < clt 8867    <_ cle 8868
This theorem is referenced by:  numlt  10143  bposlem8  20530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-addrcl 8798  ax-pre-lttri 8811  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873
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