ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  decsubi Unicode version

Theorem decsubi 9212
Description: Difference between a numeral  M and a nonnegative integer  N (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
Hypotheses
Ref Expression
decaddi.1  |-  A  e. 
NN0
decaddi.2  |-  B  e. 
NN0
decaddi.3  |-  N  e. 
NN0
decaddi.4  |-  M  = ; A B
decaddci.5  |-  ( A  +  1 )  =  D
decsubi.5  |-  ( B  -  N )  =  C
Assertion
Ref Expression
decsubi  |-  ( M  -  N )  = ; A C

Proof of Theorem decsubi
StepHypRef Expression
1 10nn0 9167 . . . . 5  |- ; 1 0  e.  NN0
2 decaddi.1 . . . . 5  |-  A  e. 
NN0
31, 2nn0mulcli 8983 . . . 4  |-  (; 1 0  x.  A
)  e.  NN0
43nn0cni 8957 . . 3  |-  (; 1 0  x.  A
)  e.  CC
5 decaddi.2 . . . 4  |-  B  e. 
NN0
65nn0cni 8957 . . 3  |-  B  e.  CC
7 decaddi.3 . . . 4  |-  N  e. 
NN0
87nn0cni 8957 . . 3  |-  N  e.  CC
94, 6, 8addsubassi 8021 . 2  |-  ( ( (; 1 0  x.  A
)  +  B )  -  N )  =  ( (; 1 0  x.  A
)  +  ( B  -  N ) )
10 decaddi.4 . . . 4  |-  M  = ; A B
11 dfdec10 9153 . . . 4  |- ; A B  =  ( (; 1 0  x.  A
)  +  B )
1210, 11eqtri 2138 . . 3  |-  M  =  ( (; 1 0  x.  A
)  +  B )
1312oveq1i 5752 . 2  |-  ( M  -  N )  =  ( ( (; 1 0  x.  A
)  +  B )  -  N )
14 dfdec10 9153 . . 3  |- ; A C  =  ( (; 1 0  x.  A
)  +  C )
15 decsubi.5 . . . . 5  |-  ( B  -  N )  =  C
1615eqcomi 2121 . . . 4  |-  C  =  ( B  -  N
)
1716oveq2i 5753 . . 3  |-  ( (; 1
0  x.  A )  +  C )  =  ( (; 1 0  x.  A
)  +  ( B  -  N ) )
1814, 17eqtri 2138 . 2  |- ; A C  =  ( (; 1 0  x.  A
)  +  ( B  -  N ) )
199, 13, 183eqtr4i 2148 1  |-  ( M  -  N )  = ; A C
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465  (class class class)co 5742   0cc0 7588   1c1 7589    + caddc 7591    x. cmul 7593    - cmin 7901   NN0cn0 8945  ;cdc 9150
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sub 7903  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-5 8750  df-6 8751  df-7 8752  df-8 8753  df-9 8754  df-n0 8946  df-dec 9151
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator