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Theorem numltc 10160
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 10159 . . . 4  |-  ( ( T  x.  A )  +  C )  < 
( ( T  x.  A )  +  T
)
61nnrei 9771 . . . . . . 7  |-  T  e.  RR
76recni 8865 . . . . . 6  |-  T  e.  CC
82nn0rei 9992 . . . . . . 7  |-  A  e.  RR
98recni 8865 . . . . . 6  |-  A  e.  CC
10 ax-1cn 8811 . . . . . 6  |-  1  e.  CC
117, 9, 10adddii 8863 . . . . 5  |-  ( T  x.  ( A  + 
1 ) )  =  ( ( T  x.  A )  +  ( T  x.  1 ) )
127mulid1i 8855 . . . . . 6  |-  ( T  x.  1 )  =  T
1312oveq2i 5885 . . . . 5  |-  ( ( T  x.  A )  +  ( T  x.  1 ) )  =  ( ( T  x.  A )  +  T
)
1411, 13eqtri 2316 . . . 4  |-  ( T  x.  ( A  + 
1 ) )  =  ( ( T  x.  A )  +  T
)
155, 14breqtrri 4064 . . 3  |-  ( ( T  x.  A )  +  C )  < 
( T  x.  ( A  +  1 ) )
16 numltc.6 . . . . 5  |-  A  < 
B
17 numlt.3 . . . . . 6  |-  B  e. 
NN0
18 nn0ltp1le 10090 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A  <  B  <->  ( A  +  1 )  <_  B ) )
192, 17, 18mp2an 653 . . . . 5  |-  ( A  <  B  <->  ( A  +  1 )  <_  B )
2016, 19mpbi 199 . . . 4  |-  ( A  +  1 )  <_  B
211nngt0i 9795 . . . . 5  |-  0  <  T
22 peano2re 9001 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
238, 22ax-mp 8 . . . . . 6  |-  ( A  +  1 )  e.  RR
2417nn0rei 9992 . . . . . 6  |-  B  e.  RR
2523, 24, 6lemul2i 9696 . . . . 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 199 . . 3  |-  ( T  x.  ( A  + 
1 ) )  <_ 
( T  x.  B
)
286, 8remulcli 8867 . . . . 5  |-  ( T  x.  A )  e.  RR
293nn0rei 9992 . . . . 5  |-  C  e.  RR
3028, 29readdcli 8866 . . . 4  |-  ( ( T  x.  A )  +  C )  e.  RR
316, 23remulcli 8867 . . . 4  |-  ( T  x.  ( A  + 
1 ) )  e.  RR
326, 24remulcli 8867 . . . 4  |-  ( T  x.  B )  e.  RR
3330, 31, 32ltletri 8963 . . 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 653 . 2  |-  ( ( T  x.  A )  +  C )  < 
( T  x.  B
)
35 numltc.4 . . 3  |-  D  e. 
NN0
3632, 35nn0addge1i 10028 . 2  |-  ( T  x.  B )  <_ 
( ( T  x.  B )  +  D
)
3735nn0rei 9992 . . . 4  |-  D  e.  RR
3832, 37readdcli 8866 . . 3  |-  ( ( T  x.  B )  +  D )  e.  RR
3930, 32, 38ltletri 8963 . 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 653 1  |-  ( ( T  x.  A )  +  C )  < 
( ( T  x.  B )  +  D
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884   NNcn 9762   NN0cn0 9981
This theorem is referenced by:  decltc  10162  numlti  10164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041
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