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Theorem numltc 9219
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 9218 . . . 4  |-  ( ( T  x.  A )  +  C )  < 
( ( T  x.  A )  +  T
)
61nnrei 8741 . . . . . . 7  |-  T  e.  RR
76recni 7790 . . . . . 6  |-  T  e.  CC
82nn0rei 9000 . . . . . . 7  |-  A  e.  RR
98recni 7790 . . . . . 6  |-  A  e.  CC
10 ax-1cn 7725 . . . . . 6  |-  1  e.  CC
117, 9, 10adddii 7788 . . . . 5  |-  ( T  x.  ( A  + 
1 ) )  =  ( ( T  x.  A )  +  ( T  x.  1 ) )
127mulid1i 7780 . . . . . 6  |-  ( T  x.  1 )  =  T
1312oveq2i 5785 . . . . 5  |-  ( ( T  x.  A )  +  ( T  x.  1 ) )  =  ( ( T  x.  A )  +  T
)
1411, 13eqtri 2160 . . . 4  |-  ( T  x.  ( A  + 
1 ) )  =  ( ( T  x.  A )  +  T
)
155, 14breqtrri 3955 . . 3  |-  ( ( T  x.  A )  +  C )  < 
( T  x.  ( A  +  1 ) )
16 numltc.6 . . . . 5  |-  A  < 
B
17 numlt.3 . . . . . 6  |-  B  e. 
NN0
18 nn0ltp1le 9128 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A  <  B  <->  ( A  +  1 )  <_  B ) )
192, 17, 18mp2an 422 . . . . 5  |-  ( A  <  B  <->  ( A  +  1 )  <_  B )
2016, 19mpbi 144 . . . 4  |-  ( A  +  1 )  <_  B
211nngt0i 8762 . . . . 5  |-  0  <  T
22 peano2re 7910 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
238, 22ax-mp 5 . . . . . 6  |-  ( A  +  1 )  e.  RR
2417nn0rei 9000 . . . . . 6  |-  B  e.  RR
2523, 24, 6lemul2i 8695 . . . . 5  |-  ( 0  <  T  ->  (
( A  +  1 )  <_  B  <->  ( T  x.  ( A  +  1 ) )  <_  ( T  x.  B )
) )
2621, 25ax-mp 5 . . . 4  |-  ( ( A  +  1 )  <_  B  <->  ( T  x.  ( A  +  1 ) )  <_  ( T  x.  B )
)
2720, 26mpbi 144 . . 3  |-  ( T  x.  ( A  + 
1 ) )  <_ 
( T  x.  B
)
286, 8remulcli 7792 . . . . 5  |-  ( T  x.  A )  e.  RR
293nn0rei 9000 . . . . 5  |-  C  e.  RR
3028, 29readdcli 7791 . . . 4  |-  ( ( T  x.  A )  +  C )  e.  RR
316, 23remulcli 7792 . . . 4  |-  ( T  x.  ( A  + 
1 ) )  e.  RR
326, 24remulcli 7792 . . . 4  |-  ( T  x.  B )  e.  RR
3330, 31, 32ltletri 7882 . . 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 422 . 2  |-  ( ( T  x.  A )  +  C )  < 
( T  x.  B
)
35 numltc.4 . . 3  |-  D  e. 
NN0
3632, 35nn0addge1i 9037 . 2  |-  ( T  x.  B )  <_ 
( ( T  x.  B )  +  D
)
3735nn0rei 9000 . . . 4  |-  D  e.  RR
3832, 37readdcli 7791 . . 3  |-  ( ( T  x.  B )  +  D )  e.  RR
3930, 32, 38ltletri 7882 . 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 422 1  |-  ( ( T  x.  A )  +  C )  < 
( ( T  x.  B )  +  D
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   RRcr 7631   0cc0 7632   1c1 7633    + caddc 7635    x. cmul 7637    < clt 7812    <_ cle 7813   NNcn 8732   NN0cn0 8989
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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-ltadd 7748  ax-pre-mulgt0 7749
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067
This theorem is referenced by:  decltc  9222  numlti  9230
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