Theorem decsplit 12623

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 9491	. . . . . 6 |- ; 1 0 e. NN0
2		decsplit0.1	. . . . . . 7 |- A e. NN0
3		decsplit.4	. . . . . . . 8 |- M e. NN0
4	1, 3	nn0expcli 10674	. . . . . . 7 |- (; 1 0 ^ M ) e. NN0
5	2, 4	nn0mulcli 9304	. . . . . 6 |- ( A x. (; 1 0 ^ M ) ) e. NN0
6	1, 5	nn0mulcli 9304	. . . . 5 |- (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) e. NN0
7	6	nn0cni 9278	. . . 4 |- (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) e. CC
8		decsplit.2	. . . . . 6 |- B e. NN0
9	1, 8	nn0mulcli 9304	. . . . 5 |- (; 1 0 x. B ) e. NN0
10	9	nn0cni 9278	. . . 4 |- (; 1 0 x. B ) e. CC
11		decsplit.3	. . . . 5 |- D e. NN0
12	11	nn0cni 9278	. . . 4 |- D e. CC
13	7, 10, 12	addassi 8051	. . 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 9278	. . . . . 6 |- ; 1 0 e. CC
15	5	nn0cni 9278	. . . . . 6 |- ( A x. (; 1 0 ^ M ) ) e. CC
16	8	nn0cni 9278	. . . . . 6 |- B e. CC
17	14, 15, 16	adddii 8053	. . . . 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 5936	. . . . 5 |- (; 1 0 x. ( ( A x. (; 1 0 ^ M ) ) + B ) ) = (; 1 0 x. C )
20	17, 19	eqtr3i 2219	. . . 4 |- ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + (; 1 0 x. B ) ) = (; 1 0 x. C )
21	20	oveq1i 5935	. . 3 |- ( ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + (; 1 0 x. B ) ) + D ) = ( (; 1 0 x. C ) + D )
22	13, 21	eqtr3i 2219	. 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 9278	. . . . . . 7 |- (; 1 0 ^ M ) e. CC
25	24, 14	mulcomi 8049	. . . . . 6 |- ( (; 1 0 ^ M ) x. ; 1 0 ) = (; 1 0 x. (; 1 0 ^ M ) )
26	1, 3, 23, 25	numexpp1 12618	. . . . 5 |- (; 1 0 ^ N ) = (; 1 0 x. (; 1 0 ^ M ) )
27	26	oveq2i 5936	. . . 4 |- ( A x. (; 1 0 ^ N ) ) = ( A x. (; 1 0 x. (; 1 0 ^ M ) ) )
28	2	nn0cni 9278	. . . . 5 |- A e. CC
29	28, 14, 24	mul12i 8189	. . . 4 |- ( A x. (; 1 0 x. (; 1 0 ^ M ) ) ) = (; 1 0 x. ( A x. (; 1 0 ^ M ) ) )
30	27, 29	eqtri 2217	. . 3 |- ( A x. (; 1 0 ^ N ) ) = (; 1 0 x. ( A x. (; 1 0 ^ M ) ) )
31		dfdec10 9477	. . 3 |- ; B D = ( (; 1 0 x. B ) + D )
32	30, 31	oveq12i 5937	. 2 |- ( ( A x. (; 1 0 ^ N ) ) + ; B D ) = ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + ( (; 1 0 x. B ) + D ) )
33		dfdec10 9477	. 2 |- ; C D = ( (; 1 0 x. C ) + D )
34	22, 32, 33	3eqtr4i 2227	1 |- ( ( A x. (; 1 0 ^ N ) ) + ; B D ) = ; C D

Syntax hints:   = wceq 1364   e. wcel 2167  (class class class)co 5925   0cc0 7896   1c1 7897   + caddc 7899   x. cmul 7901   NN0cn0 9266  ;cdc 9474   ^cexp 10647

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-9 9073  df-n0 9267  df-z 9344  df-dec 9475  df-uz 9619  df-seqfrec 10557  df-exp 10648

This theorem is referenced by: (None)