Theorem decsplit 13210

Description: Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)

Hypotheses

Ref	Expression
decsplit0.1	|- A e. NN0
decsplit.2	|- B e. NN0
decsplit.3	|- D e. NN0
decsplit.4	|- M e. NN0
decsplit.5	|- ( M + 1 ) = N
decsplit.6	|- ( ( A x. (; 1 0 ^ M ) ) + B ) = C

Assertion

Ref	Expression
decsplit	|- ( ( A x. (; 1 0 ^ N ) ) + ; B D ) = ; C D

Proof of Theorem decsplit

Step	Hyp	Ref	Expression
1		10nn0 9796	. . . . . 6 |- ; 1 0 e. NN0
2		decsplit0.1	. . . . . . 7 |- A e. NN0
3		decsplit.4	. . . . . . . 8 |- M e. NN0
4	1, 3	nn0expcli 11004	. . . . . . 7 |- (; 1 0 ^ M ) e. NN0
5	2, 4	nn0mulcli 9603	. . . . . 6 |- ( A x. (; 1 0 ^ M ) ) e. NN0
6	1, 5	nn0mulcli 9603	. . . . 5 |- (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) e. NN0
7	6	nn0cni 9577	. . . 4 |- (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) e. CC
8		decsplit.2	. . . . . 6 |- B e. NN0
9	1, 8	nn0mulcli 9603	. . . . 5 |- (; 1 0 x. B ) e. NN0
10	9	nn0cni 9577	. . . 4 |- (; 1 0 x. B ) e. CC
11		decsplit.3	. . . . 5 |- D e. NN0
12	11	nn0cni 9577	. . . 4 |- D e. CC
13	7, 10, 12	addassi 8334	. . 3 |- ( ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + (; 1 0 x. B ) ) + D ) = ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + ( (; 1 0 x. B ) + D ) )
14	1	nn0cni 9577	. . . . . 6 |- ; 1 0 e. CC
15	5	nn0cni 9577	. . . . . 6 |- ( A x. (; 1 0 ^ M ) ) e. CC
16	8	nn0cni 9577	. . . . . 6 |- B e. CC
17	14, 15, 16	adddii 8336	. . . . 5 |- (; 1 0 x. ( ( A x. (; 1 0 ^ M ) ) + B ) ) = ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + (; 1 0 x. B ) )
18		decsplit.6	. . . . . 6 |- ( ( A x. (; 1 0 ^ M ) ) + B ) = C
19	18	oveq2i 6096	. . . . 5 |- (; 1 0 x. ( ( A x. (; 1 0 ^ M ) ) + B ) ) = (; 1 0 x. C )
20	17, 19	eqtr3i 2261	. . . 4 |- ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + (; 1 0 x. B ) ) = (; 1 0 x. C )
21	20	oveq1i 6095	. . 3 |- ( ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + (; 1 0 x. B ) ) + D ) = ( (; 1 0 x. C ) + D )
22	13, 21	eqtr3i 2261	. 2 |- ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + ( (; 1 0 x. B ) + D ) ) = ( (; 1 0 x. C ) + D )
23		decsplit.5	. . . . . 6 |- ( M + 1 ) = N
24	4	nn0cni 9577	. . . . . . 7 |- (; 1 0 ^ M ) e. CC
25	24, 14	mulcomi 8332	. . . . . 6 |- ( (; 1 0 ^ M ) x. ; 1 0 ) = (; 1 0 x. (; 1 0 ^ M ) )
26	1, 3, 23, 25	numexpp1 13205	. . . . 5 |- (; 1 0 ^ N ) = (; 1 0 x. (; 1 0 ^ M ) )
27	26	oveq2i 6096	. . . 4 |- ( A x. (; 1 0 ^ N ) ) = ( A x. (; 1 0 x. (; 1 0 ^ M ) ) )
28	2	nn0cni 9577	. . . . 5 |- A e. CC
29	28, 14, 24	mul12i 8472	. . . 4 |- ( A x. (; 1 0 x. (; 1 0 ^ M ) ) ) = (; 1 0 x. ( A x. (; 1 0 ^ M ) ) )
30	27, 29	eqtri 2259	. . 3 |- ( A x. (; 1 0 ^ N ) ) = (; 1 0 x. ( A x. (; 1 0 ^ M ) ) )
31		dfdec10 9782	. . 3 |- ; B D = ( (; 1 0 x. B ) + D )
32	30, 31	oveq12i 6097	. 2 |- ( ( A x. (; 1 0 ^ N ) ) + ; B D ) = ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + ( (; 1 0 x. B ) + D ) )
33		dfdec10 9782	. 2 |- ; C D = ( (; 1 0 x. C ) + D )
34	22, 32, 33	3eqtr4i 2269	1 |- ( ( A x. (; 1 0 ^ N ) ) + ; B D ) = ; C D

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6085   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   NN0cn0 9565  ;cdc 9779   ^cexp 10977

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-z 9647  df-dec 9780  df-uz 9924  df-seqfrec 10887  df-exp 10978

This theorem is used by: (None)