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Theorem numltc 10335
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 10334 . . . 4  |-  ( ( T  x.  A )  +  C )  < 
( ( T  x.  A )  +  T
)
61nnrei 9942 . . . . . . 7  |-  T  e.  RR
76recni 9036 . . . . . 6  |-  T  e.  CC
82nn0rei 10165 . . . . . . 7  |-  A  e.  RR
98recni 9036 . . . . . 6  |-  A  e.  CC
10 ax-1cn 8982 . . . . . 6  |-  1  e.  CC
117, 9, 10adddii 9034 . . . . 5  |-  ( T  x.  ( A  + 
1 ) )  =  ( ( T  x.  A )  +  ( T  x.  1 ) )
127mulid1i 9026 . . . . . 6  |-  ( T  x.  1 )  =  T
1312oveq2i 6032 . . . . 5  |-  ( ( T  x.  A )  +  ( T  x.  1 ) )  =  ( ( T  x.  A )  +  T
)
1411, 13eqtri 2408 . . . 4  |-  ( T  x.  ( A  + 
1 ) )  =  ( ( T  x.  A )  +  T
)
155, 14breqtrri 4179 . . 3  |-  ( ( T  x.  A )  +  C )  < 
( T  x.  ( A  +  1 ) )
16 numltc.6 . . . . 5  |-  A  < 
B
17 numlt.3 . . . . . 6  |-  B  e. 
NN0
18 nn0ltp1le 10265 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A  <  B  <->  ( A  +  1 )  <_  B ) )
192, 17, 18mp2an 654 . . . . 5  |-  ( A  <  B  <->  ( A  +  1 )  <_  B )
2016, 19mpbi 200 . . . 4  |-  ( A  +  1 )  <_  B
211nngt0i 9966 . . . . 5  |-  0  <  T
22 peano2re 9172 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
238, 22ax-mp 8 . . . . . 6  |-  ( A  +  1 )  e.  RR
2417nn0rei 10165 . . . . . 6  |-  B  e.  RR
2523, 24, 6lemul2i 9867 . . . . 5  |-  ( 0  <  T  ->  (
( A  +  1 )  <_  B  <->  ( T  x.  ( A  +  1 ) )  <_  ( T  x.  B )
) )
2621, 25ax-mp 8 . . . 4  |-  ( ( A  +  1 )  <_  B  <->  ( T  x.  ( A  +  1 ) )  <_  ( T  x.  B )
)
2720, 26mpbi 200 . . 3  |-  ( T  x.  ( A  + 
1 ) )  <_ 
( T  x.  B
)
286, 8remulcli 9038 . . . . 5  |-  ( T  x.  A )  e.  RR
293nn0rei 10165 . . . . 5  |-  C  e.  RR
3028, 29readdcli 9037 . . . 4  |-  ( ( T  x.  A )  +  C )  e.  RR
316, 23remulcli 9038 . . . 4  |-  ( T  x.  ( A  + 
1 ) )  e.  RR
326, 24remulcli 9038 . . . 4  |-  ( T  x.  B )  e.  RR
3330, 31, 32ltletri 9134 . . 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 654 . 2  |-  ( ( T  x.  A )  +  C )  < 
( T  x.  B
)
35 numltc.4 . . 3  |-  D  e. 
NN0
3632, 35nn0addge1i 10201 . 2  |-  ( T  x.  B )  <_ 
( ( T  x.  B )  +  D
)
3735nn0rei 10165 . . . 4  |-  D  e.  RR
3832, 37readdcli 9037 . . 3  |-  ( ( T  x.  B )  +  D )  e.  RR
3930, 32, 38ltletri 9134 . 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 654 1  |-  ( ( T  x.  A )  +  C )  < 
( ( T  x.  B )  +  D
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1717   class class class wbr 4154  (class class class)co 6021   RRcr 8923   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    < clt 9054    <_ cle 9055   NNcn 9933   NN0cn0 10154
This theorem is referenced by:  decltc  10337  numlti  10339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216
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