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Theorem numsucc 9399
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 9168 . . . . . . . 8  |-  1  e.  NN0
42, 3nn0addcli 9189 . . . . . . 7  |-  ( Y  +  1 )  e. 
NN0
51, 4eqeltri 2250 . . . . . 6  |-  T  e. 
NN0
65nn0cni 9164 . . . . 5  |-  T  e.  CC
76mulid1i 7937 . . . 4  |-  ( T  x.  1 )  =  T
87oveq2i 5879 . . 3  |-  ( ( T  x.  A )  +  ( T  x.  1 ) )  =  ( ( T  x.  A )  +  T
)
9 numsucc.3 . . . . 5  |-  A  e. 
NN0
109nn0cni 9164 . . . 4  |-  A  e.  CC
11 ax-1cn 7882 . . . 4  |-  1  e.  CC
126, 10, 11adddii 7945 . . 3  |-  ( T  x.  ( A  + 
1 ) )  =  ( ( T  x.  A )  +  ( T  x.  1 ) )
131eqcomi 2181 . . . 4  |-  ( Y  +  1 )  =  T
14 numsucc.5 . . . 4  |-  N  =  ( ( T  x.  A )  +  Y
)
155, 9, 2, 13, 14numsuc 9373 . . 3  |-  ( N  +  1 )  =  ( ( T  x.  A )  +  T
)
168, 12, 153eqtr4ri 2209 . 2  |-  ( N  +  1 )  =  ( T  x.  ( A  +  1 ) )
17 numsucc.4 . . 3  |-  ( A  +  1 )  =  B
1817oveq2i 5879 . 2  |-  ( T  x.  ( A  + 
1 ) )  =  ( T  x.  B
)
199, 3nn0addcli 9189 . . . 4  |-  ( A  +  1 )  e. 
NN0
2017, 19eqeltrri 2251 . . 3  |-  B  e. 
NN0
215, 20num0u 9370 . 2  |-  ( T  x.  B )  =  ( ( T  x.  B )  +  0 )
2216, 18, 213eqtri 2202 1  |-  ( N  +  1 )  =  ( ( T  x.  B )  +  0 )
Colors of variables: wff set class
Syntax hints:    = wceq 1353    e. wcel 2148  (class class class)co 5868   0cc0 7789   1c1 7790    + caddc 7792    x. cmul 7794   NN0cn0 9152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-sub 8107  df-inn 8896  df-n0 9153
This theorem is referenced by:  decsucc  9400
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