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Theorem ltadd2 8174
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 axltadd 7827 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) ) )
2 ax-rnegex 7722 . . . 4  |-  ( C  e.  RR  ->  E. x  e.  RR  ( C  +  x )  =  0 )
323ad2ant3 1004 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  E. x  e.  RR  ( C  +  x )  =  0 )
4 simpl3 986 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  RR )
5 simpl1 984 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  RR )
64, 5readdcld 7788 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( C  +  A )  e.  RR )
7 simpl2 985 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  RR )
84, 7readdcld 7788 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( C  +  B )  e.  RR )
9 simprl 520 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  RR )
10 axltadd 7827 . . . . . 6  |-  ( ( ( C  +  A
)  e.  RR  /\  ( C  +  B
)  e.  RR  /\  x  e.  RR )  ->  ( ( C  +  A )  <  ( C  +  B )  ->  ( x  +  ( C  +  A ) )  <  ( x  +  ( C  +  B ) ) ) )
116, 8, 9, 10syl3anc 1216 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( ( C  +  A )  <  ( C  +  B
)  ->  ( x  +  ( C  +  A ) )  < 
( x  +  ( C  +  B ) ) ) )
129recnd 7787 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  CC )
134recnd 7787 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  CC )
145recnd 7787 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  CC )
1512, 13, 14addassd 7781 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  A )  =  ( x  +  ( C  +  A ) ) )
167recnd 7787 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  CC )
1712, 13, 16addassd 7781 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  B )  =  ( x  +  ( C  +  B ) ) )
1815, 17breq12d 3937 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
( x  +  C
)  +  A )  <  ( ( x  +  C )  +  B )  <->  ( x  +  ( C  +  A ) )  < 
( x  +  ( C  +  B ) ) ) )
1911, 18sylibrd 168 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( ( C  +  A )  <  ( C  +  B
)  ->  ( (
x  +  C )  +  A )  < 
( ( x  +  C )  +  B
) ) )
20 simprr 521 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( C  +  x )  =  0 )
21 addcom 7892 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  x  e.  CC )  ->  ( C  +  x
)  =  ( x  +  C ) )
2221eqeq1d 2146 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  x  e.  CC )  ->  ( ( C  +  x )  =  0  <-> 
( x  +  C
)  =  0 ) )
2313, 12, 22syl2anc 408 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( ( C  +  x )  =  0  <->  ( x  +  C )  =  0 ) )
2420, 23mpbid 146 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( x  +  C )  =  0 )
2524oveq1d 5782 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  A )  =  ( 0  +  A
) )
2614addid2d 7905 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( 0  +  A )  =  A )
2725, 26eqtrd 2170 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  A )  =  A )
2824oveq1d 5782 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  B )  =  ( 0  +  B
) )
2916addid2d 7905 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( 0  +  B )  =  B )
3028, 29eqtrd 2170 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  B )  =  B )
3127, 30breq12d 3937 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
( x  +  C
)  +  A )  <  ( ( x  +  C )  +  B )  <->  A  <  B ) )
3219, 31sylibd 148 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( ( C  +  A )  <  ( C  +  B
)  ->  A  <  B ) )
333, 32rexlimddv 2552 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  ->  A  <  B ) )
341, 33impbid 128 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2415   class class class wbr 3924  (class class class)co 5767   CCcc 7611   RRcr 7612   0cc0 7613    + caddc 7616    < clt 7793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-pre-ltadd 7729
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-iota 5083  df-fv 5126  df-ov 5770  df-pnf 7795  df-mnf 7796  df-ltxr 7798
This theorem is referenced by:  ltadd2i  8175  ltadd2d  8176  ltaddneg  8179  ltadd1  8184  ltaddpos  8207  ltsub2  8214  ltaddsublt  8326  avglt1  8951  flqbi2  10057
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