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Theorem deccarry 38860
Description: Add 1 to a 2 digit number with carry. This is a special case of decsucc 11101, but in closed form. As observed by ML, this theorem allows for carrying the 1 down multiple decimal constructors, so we can carry the 1 multiple times down a multi-digit number, e.g. by applying this theorem three times we get  (;; 9 9 9  +  1 )  = ;;; 1 0 0 0. (Contributed by AV, 4-Aug-2020.) (Revised by ML, 8-Aug-2020.)
Assertion
Ref Expression
deccarry  |-  ( A  e.  NN  ->  (; A 9  +  1 )  = ; ( A  +  1 ) 0 )

Proof of Theorem deccarry
StepHypRef Expression
1 df-dec 11075 . . . 4  |- ; A 9  =  ( ( 10  x.  A
)  +  9 )
21oveq1i 6318 . . 3  |-  (; A 9  +  1 )  =  ( ( ( 10  x.  A
)  +  9 )  +  1 )
3 10nn 10798 . . . . . . 7  |-  10  e.  NN
43a1i 11 . . . . . 6  |-  ( A  e.  NN  ->  10  e.  NN )
5 id 22 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  NN )
64, 5nnmulcld 10679 . . . . 5  |-  ( A  e.  NN  ->  ( 10  x.  A )  e.  NN )
76nncnd 10647 . . . 4  |-  ( A  e.  NN  ->  ( 10  x.  A )  e.  CC )
8 9cn 10719 . . . . 5  |-  9  e.  CC
98a1i 11 . . . 4  |-  ( A  e.  NN  ->  9  e.  CC )
10 1cnd 9677 . . . 4  |-  ( A  e.  NN  ->  1  e.  CC )
117, 9, 10addassd 9683 . . 3  |-  ( A  e.  NN  ->  (
( ( 10  x.  A )  +  9 )  +  1 )  =  ( ( 10  x.  A )  +  ( 9  +  1 ) ) )
122, 11syl5eq 2517 . 2  |-  ( A  e.  NN  ->  (; A 9  +  1 )  =  ( ( 10  x.  A )  +  ( 9  +  1 ) ) )
13 9p1e10 10764 . . . 4  |-  ( 9  +  1 )  =  10
1413oveq2i 6319 . . 3  |-  ( ( 10  x.  A )  +  ( 9  +  1 ) )  =  ( ( 10  x.  A )  +  10 )
153nncni 10641 . . . . . . 7  |-  10  e.  CC
1615mulid1i 9663 . . . . . 6  |-  ( 10  x.  1 )  =  10
1716eqcomi 2480 . . . . 5  |-  10  =  ( 10  x.  1
)
1817oveq2i 6319 . . . 4  |-  ( ( 10  x.  A )  +  10 )  =  ( ( 10  x.  A )  +  ( 10  x.  1 ) )
1915a1i 11 . . . . . 6  |-  ( A  e.  NN  ->  10  e.  CC )
20 nncn 10639 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  CC )
21 adddi 9646 . . . . . . 7  |-  ( ( 10  e.  CC  /\  A  e.  CC  /\  1  e.  CC )  ->  ( 10  x.  ( A  + 
1 ) )  =  ( ( 10  x.  A )  +  ( 10  x.  1 ) ) )
2221eqcomd 2477 . . . . . 6  |-  ( ( 10  e.  CC  /\  A  e.  CC  /\  1  e.  CC )  ->  (
( 10  x.  A
)  +  ( 10  x.  1 ) )  =  ( 10  x.  ( A  +  1
) ) )
2319, 20, 10, 22syl3anc 1292 . . . . 5  |-  ( A  e.  NN  ->  (
( 10  x.  A
)  +  ( 10  x.  1 ) )  =  ( 10  x.  ( A  +  1
) ) )
24 df-dec 11075 . . . . . 6  |- ; ( A  +  1 ) 0  =  ( ( 10  x.  ( A  +  1 ) )  +  0 )
25 peano2nn 10643 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
2625nncnd 10647 . . . . . . . 8  |-  ( A  e.  NN  ->  ( A  +  1 )  e.  CC )
2719, 26mulcld 9681 . . . . . . 7  |-  ( A  e.  NN  ->  ( 10  x.  ( A  + 
1 ) )  e.  CC )
2827addid1d 9851 . . . . . 6  |-  ( A  e.  NN  ->  (
( 10  x.  ( A  +  1 ) )  +  0 )  =  ( 10  x.  ( A  +  1
) ) )
2924, 28syl5req 2518 . . . . 5  |-  ( A  e.  NN  ->  ( 10  x.  ( A  + 
1 ) )  = ; ( A  +  1 ) 0 )
3023, 29eqtrd 2505 . . . 4  |-  ( A  e.  NN  ->  (
( 10  x.  A
)  +  ( 10  x.  1 ) )  = ; ( A  +  1
) 0 )
3118, 30syl5eq 2517 . . 3  |-  ( A  e.  NN  ->  (
( 10  x.  A
)  +  10 )  = ; ( A  +  1
) 0 )
3214, 31syl5eq 2517 . 2  |-  ( A  e.  NN  ->  (
( 10  x.  A
)  +  ( 9  +  1 ) )  = ; ( A  +  1
) 0 )
3312, 32eqtrd 2505 1  |-  ( A  e.  NN  ->  (; A 9  +  1 )  = ; ( A  +  1 ) 0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 1007    = wceq 1452    e. wcel 1904  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   NNcn 10631   9c9 10688   10c10 10689  ;cdc 11074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-ltxr 9698  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-dec 11075
This theorem is referenced by: (None)
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