Theorem decsplit 13191

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 9777	. . . . . 6 |- ; 1 0 e. NN0
2		decsplit0.1	. . . . . . 7 |- A e. NN0
3		decsplit.4	. . . . . . . 8 |- M e. NN0
4	1, 3	nn0expcli 10985	. . . . . . 7 |- (; 1 0 ^ M ) e. NN0
5	2, 4	nn0mulcli 9584	. . . . . 6 |- ( A x. (; 1 0 ^ M ) ) e. NN0
6	1, 5	nn0mulcli 9584	. . . . 5 |- (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) e. NN0
7	6	nn0cni 9558	. . . 4 |- (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) e. CC
8		decsplit.2	. . . . . 6 |- B e. NN0
9	1, 8	nn0mulcli 9584	. . . . 5 |- (; 1 0 x. B ) e. NN0
10	9	nn0cni 9558	. . . 4 |- (; 1 0 x. B ) e. CC
11		decsplit.3	. . . . 5 |- D e. NN0
12	11	nn0cni 9558	. . . 4 |- D e. CC
13	7, 10, 12	addassi 8328	. . 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 9558	. . . . . 6 |- ; 1 0 e. CC
15	5	nn0cni 9558	. . . . . 6 |- ( A x. (; 1 0 ^ M ) ) e. CC
16	8	nn0cni 9558	. . . . . 6 |- B e. CC
17	14, 15, 16	adddii 8330	. . . . 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 6090	. . . . 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 6089	. . 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 9558	. . . . . . 7 |- (; 1 0 ^ M ) e. CC
25	24, 14	mulcomi 8326	. . . . . 6 |- ( (; 1 0 ^ M ) x. ; 1 0 ) = (; 1 0 x. (; 1 0 ^ M ) )
26	1, 3, 23, 25	numexpp1 13186	. . . . 5 |- (; 1 0 ^ N ) = (; 1 0 x. (; 1 0 ^ M ) )
27	26	oveq2i 6090	. . . 4 |- ( A x. (; 1 0 ^ N ) ) = ( A x. (; 1 0 x. (; 1 0 ^ M ) ) )
28	2	nn0cni 9558	. . . . 5 |- A e. CC
29	28, 14, 24	mul12i 8466	. . . 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 9763	. . 3 |- ; B D = ( (; 1 0 x. B ) + D )
32	30, 31	oveq12i 6091	. 2 |- ( ( A x. (; 1 0 ^ N ) ) + ; B D ) = ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + ( (; 1 0 x. B ) + D ) )
33		dfdec10 9763	. 2 |- ; C D = ( (; 1 0 x. C ) + D )
34	22, 32, 33	3eqtr4i 2269	1 |- ( ( A x. (; 1 0 ^ N ) ) + ; B D ) = ; C D

Syntax hints:   = wceq 1402   e. wcel 2209  (class class class)co 6079   0cc0 8173   1c1 8174   + caddc 8176   x. cmul 8178   NN0cn0 9546  ;cdc 9760   ^cexp 10958

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-dec 9761  df-uz 9905  df-seqfrec 10868  df-exp 10959

This theorem is referenced by: (None)