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Theorem ltadd2i 9706
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 9649 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) ) )
51, 2, 3, 4mp3an 1319 . 2  |-  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) )
6 axltadd 9649 . . . . . . 7  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( C  +  B )  <  ( C  +  A
) ) )
72, 1, 3, 6mp3an 1319 . . . . . 6  |-  ( B  <  A  ->  ( C  +  B )  <  ( C  +  A
) )
8 oveq2 6285 . . . . . 6  |-  ( B  =  A  ->  ( C  +  B )  =  ( C  +  A ) )
97, 8orim12i 516 . . . . 5  |-  ( ( B  <  A  \/  B  =  A )  ->  ( ( C  +  B )  <  ( C  +  A )  \/  ( C  +  B
)  =  ( C  +  A ) ) )
102, 1leloei 9692 . . . . 5  |-  ( B  <_  A  <->  ( B  <  A  \/  B  =  A ) )
113, 2readdcli 9600 . . . . . 6  |-  ( C  +  B )  e.  RR
123, 1readdcli 9600 . . . . . 6  |-  ( C  +  A )  e.  RR
1311, 12leloei 9692 . . . . 5  |-  ( ( C  +  B )  <_  ( C  +  A )  <->  ( ( C  +  B )  <  ( C  +  A
)  \/  ( C  +  B )  =  ( C  +  A
) ) )
149, 10, 133imtr4i 266 . . . 4  |-  ( B  <_  A  ->  ( C  +  B )  <_  ( C  +  A
) )
152, 1lenlti 9695 . . . 4  |-  ( B  <_  A  <->  -.  A  <  B )
1611, 12lenlti 9695 . . . 4  |-  ( ( C  +  B )  <_  ( C  +  A )  <->  -.  ( C  +  A )  <  ( C  +  B
) )
1714, 15, 163imtr3i 265 . . 3  |-  ( -.  A  <  B  ->  -.  ( C  +  A
)  <  ( C  +  B ) )
1817con4i 130 . 2  |-  ( ( C  +  A )  <  ( C  +  B )  ->  A  <  B )
195, 18impbii 188 1  |-  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1374    e. wcel 1762   class class class wbr 4442  (class class class)co 6277   RRcr 9482    + caddc 9486    < clt 9619    <_ cle 9620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-addrcl 9544  ax-pre-lttri 9557  ax-pre-ltadd 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625
This theorem is referenced by:  numlt  10986  bposlem8  23289
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