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Theorem ltadd2 7590
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 7249 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) ) )
2 ax-rnegex 7147 . . . 4  |-  ( C  e.  RR  ->  E. x  e.  RR  ( C  +  x )  =  0 )
323ad2ant3 962 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  E. x  e.  RR  ( C  +  x )  =  0 )
4 simpl3 944 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  RR )
5 simpl1 942 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  RR )
64, 5readdcld 7210 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( C  +  A )  e.  RR )
7 simpl2 943 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  RR )
84, 7readdcld 7210 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( C  +  B )  e.  RR )
9 simprl 498 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  RR )
10 axltadd 7249 . . . . . 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 1170 . . . . 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 7209 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  x  e.  CC )
134recnd 7209 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  C  e.  CC )
145recnd 7209 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  A  e.  CC )
1512, 13, 14addassd 7203 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  A )  =  ( x  +  ( C  +  A ) ) )
167recnd 7209 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  B  e.  CC )
1712, 13, 16addassd 7203 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  B )  =  ( x  +  ( C  +  B ) ) )
1815, 17breq12d 3806 . . . . 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 167 . . . 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 499 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( C  +  x )  =  0 )
21 addcom 7312 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  x  e.  CC )  ->  ( C  +  x
)  =  ( x  +  C ) )
2221eqeq1d 2090 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  x  e.  CC )  ->  ( ( C  +  x )  =  0  <-> 
( x  +  C
)  =  0 ) )
2313, 12, 22syl2anc 403 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( ( C  +  x )  =  0  <->  ( x  +  C )  =  0 ) )
2420, 23mpbid 145 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( x  +  C )  =  0 )
2524oveq1d 5558 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  A )  =  ( 0  +  A
) )
2614addid2d 7325 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( 0  +  A )  =  A )
2725, 26eqtrd 2114 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  A )  =  A )
2824oveq1d 5558 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  B )  =  ( 0  +  B
) )
2916addid2d 7325 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( 0  +  B )  =  B )
3028, 29eqtrd 2114 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( (
x  +  C )  +  B )  =  B )
3127, 30breq12d 3806 . . . 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 147 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( x  e.  RR  /\  ( C  +  x
)  =  0 ) )  ->  ( ( C  +  A )  <  ( C  +  B
)  ->  A  <  B ) )
333, 32rexlimddv 2482 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  ->  A  <  B ) )
341, 33impbid 127 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 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   E.wrex 2350   class class class wbr 3793  (class class class)co 5543   CCcc 7041   RRcr 7042   0cc0 7043    + caddc 7046    < clt 7215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-iota 4897  df-fv 4940  df-ov 5546  df-pnf 7217  df-mnf 7218  df-ltxr 7220
This theorem is referenced by:  ltadd2i  7591  ltadd2d  7592  ltaddneg  7595  ltadd1  7600  ltaddpos  7623  ltsub2  7630  ltaddsublt  7738  avglt1  8336  flqbi2  9373
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