Theorem decsplit 12938

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 9583	. . . . . 6 |- ; 1 0 e. NN0
2		decsplit0.1	. . . . . . 7 |- A e. NN0
3		decsplit.4	. . . . . . . 8 |- M e. NN0
4	1, 3	nn0expcli 10774	. . . . . . 7 |- (; 1 0 ^ M ) e. NN0
5	2, 4	nn0mulcli 9395	. . . . . 6 |- ( A x. (; 1 0 ^ M ) ) e. NN0
6	1, 5	nn0mulcli 9395	. . . . 5 |- (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) e. NN0
7	6	nn0cni 9369	. . . 4 |- (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) e. CC
8		decsplit.2	. . . . . 6 |- B e. NN0
9	1, 8	nn0mulcli 9395	. . . . 5 |- (; 1 0 x. B ) e. NN0
10	9	nn0cni 9369	. . . 4 |- (; 1 0 x. B ) e. CC
11		decsplit.3	. . . . 5 |- D e. NN0
12	11	nn0cni 9369	. . . 4 |- D e. CC
13	7, 10, 12	addassi 8142	. . 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 9369	. . . . . 6 |- ; 1 0 e. CC
15	5	nn0cni 9369	. . . . . 6 |- ( A x. (; 1 0 ^ M ) ) e. CC
16	8	nn0cni 9369	. . . . . 6 |- B e. CC
17	14, 15, 16	adddii 8144	. . . . 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 6005	. . . . 5 |- (; 1 0 x. ( ( A x. (; 1 0 ^ M ) ) + B ) ) = (; 1 0 x. C )
20	17, 19	eqtr3i 2252	. . . 4 |- ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + (; 1 0 x. B ) ) = (; 1 0 x. C )
21	20	oveq1i 6004	. . 3 |- ( ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + (; 1 0 x. B ) ) + D ) = ( (; 1 0 x. C ) + D )
22	13, 21	eqtr3i 2252	. 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 9369	. . . . . . 7 |- (; 1 0 ^ M ) e. CC
25	24, 14	mulcomi 8140	. . . . . 6 |- ( (; 1 0 ^ M ) x. ; 1 0 ) = (; 1 0 x. (; 1 0 ^ M ) )
26	1, 3, 23, 25	numexpp1 12933	. . . . 5 |- (; 1 0 ^ N ) = (; 1 0 x. (; 1 0 ^ M ) )
27	26	oveq2i 6005	. . . 4 |- ( A x. (; 1 0 ^ N ) ) = ( A x. (; 1 0 x. (; 1 0 ^ M ) ) )
28	2	nn0cni 9369	. . . . 5 |- A e. CC
29	28, 14, 24	mul12i 8280	. . . 4 |- ( A x. (; 1 0 x. (; 1 0 ^ M ) ) ) = (; 1 0 x. ( A x. (; 1 0 ^ M ) ) )
30	27, 29	eqtri 2250	. . 3 |- ( A x. (; 1 0 ^ N ) ) = (; 1 0 x. ( A x. (; 1 0 ^ M ) ) )
31		dfdec10 9569	. . 3 |- ; B D = ( (; 1 0 x. B ) + D )
32	30, 31	oveq12i 6006	. 2 |- ( ( A x. (; 1 0 ^ N ) ) + ; B D ) = ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + ( (; 1 0 x. B ) + D ) )
33		dfdec10 9569	. 2 |- ; C D = ( (; 1 0 x. C ) + D )
34	22, 32, 33	3eqtr4i 2260	1 |- ( ( A x. (; 1 0 ^ N ) ) + ; B D ) = ; C D

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 5994   0cc0 7987   1c1 7988   + caddc 7990   x. cmul 7992   NN0cn0 9357  ;cdc 9566   ^cexp 10747

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-7 9162  df-8 9163  df-9 9164  df-n0 9358  df-z 9435  df-dec 9567  df-uz 9711  df-seqfrec 10657  df-exp 10748

This theorem is referenced by: (None)