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Theorem numltc 8583
Description: Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
Hypotheses
Ref Expression
numlt.1  |-  T  e.  NN
numlt.2  |-  A  e. 
NN0
numlt.3  |-  B  e. 
NN0
numltc.3  |-  C  e. 
NN0
numltc.4  |-  D  e. 
NN0
numltc.5  |-  C  < 
T
numltc.6  |-  A  < 
B
Assertion
Ref Expression
numltc  |-  ( ( T  x.  A )  +  C )  < 
( ( T  x.  B )  +  D
)

Proof of Theorem numltc
StepHypRef Expression
1 numlt.1 . . . . 5  |-  T  e.  NN
2 numlt.2 . . . . 5  |-  A  e. 
NN0
3 numltc.3 . . . . 5  |-  C  e. 
NN0
4 numltc.5 . . . . 5  |-  C  < 
T
51, 2, 3, 1, 4numlt 8582 . . . 4  |-  ( ( T  x.  A )  +  C )  < 
( ( T  x.  A )  +  T
)
61nnrei 8115 . . . . . . 7  |-  T  e.  RR
76recni 7193 . . . . . 6  |-  T  e.  CC
82nn0rei 8366 . . . . . . 7  |-  A  e.  RR
98recni 7193 . . . . . 6  |-  A  e.  CC
10 ax-1cn 7131 . . . . . 6  |-  1  e.  CC
117, 9, 10adddii 7191 . . . . 5  |-  ( T  x.  ( A  + 
1 ) )  =  ( ( T  x.  A )  +  ( T  x.  1 ) )
127mulid1i 7183 . . . . . 6  |-  ( T  x.  1 )  =  T
1312oveq2i 5554 . . . . 5  |-  ( ( T  x.  A )  +  ( T  x.  1 ) )  =  ( ( T  x.  A )  +  T
)
1411, 13eqtri 2102 . . . 4  |-  ( T  x.  ( A  + 
1 ) )  =  ( ( T  x.  A )  +  T
)
155, 14breqtrri 3818 . . 3  |-  ( ( T  x.  A )  +  C )  < 
( T  x.  ( A  +  1 ) )
16 numltc.6 . . . . 5  |-  A  < 
B
17 numlt.3 . . . . . 6  |-  B  e. 
NN0
18 nn0ltp1le 8494 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A  <  B  <->  ( A  +  1 )  <_  B ) )
192, 17, 18mp2an 417 . . . . 5  |-  ( A  <  B  <->  ( A  +  1 )  <_  B )
2016, 19mpbi 143 . . . 4  |-  ( A  +  1 )  <_  B
211nngt0i 8136 . . . . 5  |-  0  <  T
22 peano2re 7311 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
238, 22ax-mp 7 . . . . . 6  |-  ( A  +  1 )  e.  RR
2417nn0rei 8366 . . . . . 6  |-  B  e.  RR
2523, 24, 6lemul2i 8070 . . . . 5  |-  ( 0  <  T  ->  (
( A  +  1 )  <_  B  <->  ( T  x.  ( A  +  1 ) )  <_  ( T  x.  B )
) )
2621, 25ax-mp 7 . . . 4  |-  ( ( A  +  1 )  <_  B  <->  ( T  x.  ( A  +  1 ) )  <_  ( T  x.  B )
)
2720, 26mpbi 143 . . 3  |-  ( T  x.  ( A  + 
1 ) )  <_ 
( T  x.  B
)
286, 8remulcli 7195 . . . . 5  |-  ( T  x.  A )  e.  RR
293nn0rei 8366 . . . . 5  |-  C  e.  RR
3028, 29readdcli 7194 . . . 4  |-  ( ( T  x.  A )  +  C )  e.  RR
316, 23remulcli 7195 . . . 4  |-  ( T  x.  ( A  + 
1 ) )  e.  RR
326, 24remulcli 7195 . . . 4  |-  ( T  x.  B )  e.  RR
3330, 31, 32ltletri 7284 . . 3  |-  ( ( ( ( T  x.  A )  +  C
)  <  ( T  x.  ( A  +  1 ) )  /\  ( T  x.  ( A  +  1 ) )  <_  ( T  x.  B ) )  -> 
( ( T  x.  A )  +  C
)  <  ( T  x.  B ) )
3415, 27, 33mp2an 417 . 2  |-  ( ( T  x.  A )  +  C )  < 
( T  x.  B
)
35 numltc.4 . . 3  |-  D  e. 
NN0
3632, 35nn0addge1i 8403 . 2  |-  ( T  x.  B )  <_ 
( ( T  x.  B )  +  D
)
3735nn0rei 8366 . . . 4  |-  D  e.  RR
3832, 37readdcli 7194 . . 3  |-  ( ( T  x.  B )  +  D )  e.  RR
3930, 32, 38ltletri 7284 . 2  |-  ( ( ( ( T  x.  A )  +  C
)  <  ( T  x.  B )  /\  ( T  x.  B )  <_  ( ( T  x.  B )  +  D
) )  ->  (
( T  x.  A
)  +  C )  <  ( ( T  x.  B )  +  D ) )
4034, 36, 39mp2an 417 1  |-  ( ( T  x.  A )  +  C )  < 
( ( T  x.  B )  +  D
)
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   RRcr 7042   0cc0 7043   1c1 7044    + caddc 7046    x. cmul 7048    < clt 7215    <_ cle 7216   NNcn 8106   NN0cn0 8355
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-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154  ax-pre-mulgt0 7155
This theorem depends on definitions:  df-bi 115  df-3or 921  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-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  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-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433
This theorem is referenced by:  decltc  8586  numlti  8594
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