Theorem decsplit 13239

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 9810	. . . . . 6 |- ; 1 0 e. NN0
2		decsplit0.1	. . . . . . 7 |- A e. NN0
3		decsplit.4	. . . . . . . 8 |- M e. NN0
4	1, 3	nn0expcli 11024	. . . . . . 7 |- (; 1 0 ^ M ) e. NN0
5	2, 4	nn0mulcli 9613	. . . . . 6 |- ( A x. (; 1 0 ^ M ) ) e. NN0
6	1, 5	nn0mulcli 9613	. . . . 5 |- (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) e. NN0
7	6	nn0cni 9587	. . . 4 |- (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) e. CC
8		decsplit.2	. . . . . 6 |- B e. NN0
9	1, 8	nn0mulcli 9613	. . . . 5 |- (; 1 0 x. B ) e. NN0
10	9	nn0cni 9587	. . . 4 |- (; 1 0 x. B ) e. CC
11		decsplit.3	. . . . 5 |- D e. NN0
12	11	nn0cni 9587	. . . 4 |- D e. CC
13	7, 10, 12	addassi 8341	. . 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 9587	. . . . . 6 |- ; 1 0 e. CC
15	5	nn0cni 9587	. . . . . 6 |- ( A x. (; 1 0 ^ M ) ) e. CC
16	8	nn0cni 9587	. . . . . 6 |- B e. CC
17	14, 15, 16	adddii 8343	. . . . 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 6099	. . . . 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 6098	. . 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 9587	. . . . . . 7 |- (; 1 0 ^ M ) e. CC
25	24, 14	mulcomi 8339	. . . . . 6 |- ( (; 1 0 ^ M ) x. ; 1 0 ) = (; 1 0 x. (; 1 0 ^ M ) )
26	1, 3, 23, 25	numexpp1 13234	. . . . 5 |- (; 1 0 ^ N ) = (; 1 0 x. (; 1 0 ^ M ) )
27	26	oveq2i 6099	. . . 4 |- ( A x. (; 1 0 ^ N ) ) = ( A x. (; 1 0 x. (; 1 0 ^ M ) ) )
28	2	nn0cni 9587	. . . . 5 |- A e. CC
29	28, 14, 24	mul12i 8480	. . . 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 9792	. . 3 |- ; B D = ( (; 1 0 x. B ) + D )
32	30, 31	oveq12i 6100	. 2 |- ( ( A x. (; 1 0 ^ N ) ) + ; B D ) = ( (; 1 0 x. ( A x. (; 1 0 ^ M ) ) ) + ( (; 1 0 x. B ) + D ) )
33		dfdec10 9792	. 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 6088   0cc0 8186   1c1 8187   + caddc 8189   x. cmul 8191   NN0cn0 9575  ;cdc 9789   ^cexp 10997

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-5 9376  df-6 9377  df-7 9378  df-8 9379  df-9 9380  df-n0 9576  df-z 9657  df-dec 9790  df-uz 9939  df-seqfrec 10907  df-exp 10998

This theorem is used by: (None)