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Theorem numsucc 8466
Description: The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
Hypotheses
Ref Expression
numsucc.1  |-  Y  e. 
NN0
numsucc.2  |-  T  =  ( Y  +  1 )
numsucc.3  |-  A  e. 
NN0
numsucc.4  |-  ( A  +  1 )  =  B
numsucc.5  |-  N  =  ( ( T  x.  A )  +  Y
)
Assertion
Ref Expression
numsucc  |-  ( N  +  1 )  =  ( ( T  x.  B )  +  0 )

Proof of Theorem numsucc
StepHypRef Expression
1 numsucc.2 . . . . . . 7  |-  T  =  ( Y  +  1 )
2 numsucc.1 . . . . . . . 8  |-  Y  e. 
NN0
3 1nn0 8255 . . . . . . . 8  |-  1  e.  NN0
42, 3nn0addcli 8276 . . . . . . 7  |-  ( Y  +  1 )  e. 
NN0
51, 4eqeltri 2126 . . . . . 6  |-  T  e. 
NN0
65nn0cni 8251 . . . . 5  |-  T  e.  CC
76mulid1i 7087 . . . 4  |-  ( T  x.  1 )  =  T
87oveq2i 5551 . . 3  |-  ( ( T  x.  A )  +  ( T  x.  1 ) )  =  ( ( T  x.  A )  +  T
)
9 numsucc.3 . . . . 5  |-  A  e. 
NN0
109nn0cni 8251 . . . 4  |-  A  e.  CC
11 ax-1cn 7035 . . . 4  |-  1  e.  CC
126, 10, 11adddii 7095 . . 3  |-  ( T  x.  ( A  + 
1 ) )  =  ( ( T  x.  A )  +  ( T  x.  1 ) )
131eqcomi 2060 . . . 4  |-  ( Y  +  1 )  =  T
14 numsucc.5 . . . 4  |-  N  =  ( ( T  x.  A )  +  Y
)
155, 9, 2, 13, 14numsuc 8440 . . 3  |-  ( N  +  1 )  =  ( ( T  x.  A )  +  T
)
168, 12, 153eqtr4ri 2087 . 2  |-  ( N  +  1 )  =  ( T  x.  ( A  +  1 ) )
17 numsucc.4 . . 3  |-  ( A  +  1 )  =  B
1817oveq2i 5551 . 2  |-  ( T  x.  ( A  + 
1 ) )  =  ( T  x.  B
)
199, 3nn0addcli 8276 . . . 4  |-  ( A  +  1 )  e. 
NN0
2017, 19eqeltrri 2127 . . 3  |-  B  e. 
NN0
215, 20num0u 8437 . 2  |-  ( T  x.  B )  =  ( ( T  x.  B )  +  0 )
2216, 18, 213eqtri 2080 1  |-  ( N  +  1 )  =  ( ( T  x.  B )  +  0 )
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409  (class class class)co 5540   0cc0 6947   1c1 6948    + caddc 6950    x. cmul 6952   NN0cn0 8239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-sub 7247  df-inn 7991  df-n0 8240
This theorem is referenced by:  decsucc  8467
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