Theorem karatsubaOLD 15793

Description: Obsolete version of karatsuba 15792 as of 9-Sep-2021. (Contributed by Mario Carneiro, 16-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
karatsubaOLD.a	|- A e. NN0
karatsubaOLD.b	|- B e. NN0
karatsubaOLD.c	|- C e. NN0
karatsubaOLD.d	|- D e. NN0
karatsubaOLD.s	|- S e. NN0
karatsubaOLD.m	|- M e. NN0
karatsubaOLD.r	|- ( A x. C ) = R
karatsubaOLD.t	|- ( B x. D ) = T
karatsubaOLD.e	|- ( ( A + B ) x. ( C + D ) ) = ( ( R + S ) + T )
karatsubaOLD.x	|- ( ( A x. ( 10 ^ M ) ) + B ) = X
karatsubaOLD.y	|- ( ( C x. ( 10 ^ M ) ) + D ) = Y
karatsubaOLD.w	|- ( ( R x. ( 10 ^ M ) ) + S ) = W
karatsubaOLD.z	|- ( ( W x. ( 10 ^ M ) ) + T ) = Z

Assertion

Ref	Expression
karatsubaOLD	|- ( X x. Y ) = Z

Proof of Theorem karatsubaOLD

Step	Hyp	Ref	Expression
1		karatsubaOLD.a	. . . . . 6 |- A e. NN0
2	1	nn0cni 11304	. . . . 5 |- A e. CC
3		10nn0OLD 11317	. . . . . . 7 |- 10 e. NN0
4	3	nn0cni 11304	. . . . . 6 |- 10 e. CC
5		karatsubaOLD.m	. . . . . 6 |- M e. NN0
6		expcl 12878	. . . . . 6 |- ( ( 10 e. CC /\ M e. NN0 ) -> ( 10 ^ M ) e. CC )
7	4, 5, 6	mp2an 708	. . . . 5 |- ( 10 ^ M ) e. CC
8	2, 7	mulcli 10045	. . . 4 |- ( A x. ( 10 ^ M ) ) e. CC
9		karatsubaOLD.b	. . . . 5 |- B e. NN0
10	9	nn0cni 11304	. . . 4 |- B e. CC
11		karatsubaOLD.c	. . . . . 6 |- C e. NN0
12	11	nn0cni 11304	. . . . 5 |- C e. CC
13	12, 7	mulcli 10045	. . . 4 |- ( C x. ( 10 ^ M ) ) e. CC
14		karatsubaOLD.d	. . . . 5 |- D e. NN0
15	14	nn0cni 11304	. . . 4 |- D e. CC
16	8, 10, 13, 15	muladdi 10481	. . 3 |- ( ( ( A x. ( 10 ^ M ) ) + B ) x. ( ( C x. ( 10 ^ M ) ) + D ) ) = ( ( ( ( A x. ( 10 ^ M ) ) x. ( C x. ( 10 ^ M ) ) ) + ( D x. B ) ) + ( ( ( A x. ( 10 ^ M ) ) x. D ) + ( ( C x. ( 10 ^ M ) ) x. B ) ) )
17	8, 13	mulcli 10045	. . . 4 |- ( ( A x. ( 10 ^ M ) ) x. ( C x. ( 10 ^ M ) ) ) e. CC
18	15, 10	mulcli 10045	. . . 4 |- ( D x. B ) e. CC
19	8, 15	mulcli 10045	. . . . 5 |- ( ( A x. ( 10 ^ M ) ) x. D ) e. CC
20	13, 10	mulcli 10045	. . . . 5 |- ( ( C x. ( 10 ^ M ) ) x. B ) e. CC
21	19, 20	addcli 10044	. . . 4 |- ( ( ( A x. ( 10 ^ M ) ) x. D ) + ( ( C x. ( 10 ^ M ) ) x. B ) ) e. CC
22	17, 18, 21	add32i 10259	. . 3 |- ( ( ( ( A x. ( 10 ^ M ) ) x. ( C x. ( 10 ^ M ) ) ) + ( D x. B ) ) + ( ( ( A x. ( 10 ^ M ) ) x. D ) + ( ( C x. ( 10 ^ M ) ) x. B ) ) ) = ( ( ( ( A x. ( 10 ^ M ) ) x. ( C x. ( 10 ^ M ) ) ) + ( ( ( A x. ( 10 ^ M ) ) x. D ) + ( ( C x. ( 10 ^ M ) ) x. B ) ) ) + ( D x. B ) )
23	8, 12	mulcli 10045	. . . . . 6 |- ( ( A x. ( 10 ^ M ) ) x. C ) e. CC
24		karatsubaOLD.s	. . . . . . 7 |- S e. NN0
25	24	nn0cni 11304	. . . . . 6 |- S e. CC
26	23, 25, 7	adddiri 10051	. . . . 5 |- ( ( ( ( A x. ( 10 ^ M ) ) x. C ) + S ) x. ( 10 ^ M ) ) = ( ( ( ( A x. ( 10 ^ M ) ) x. C ) x. ( 10 ^ M ) ) + ( S x. ( 10 ^ M ) ) )
27	2, 7, 12	mul32i 10232	. . . . . . . . 9 |- ( ( A x. ( 10 ^ M ) ) x. C ) = ( ( A x. C ) x. ( 10 ^ M ) )
28		karatsubaOLD.r	. . . . . . . . . 10 |- ( A x. C ) = R
29	28	oveq1i 6660	. . . . . . . . 9 |- ( ( A x. C ) x. ( 10 ^ M ) ) = ( R x. ( 10 ^ M ) )
30	27, 29	eqtri 2644	. . . . . . . 8 |- ( ( A x. ( 10 ^ M ) ) x. C ) = ( R x. ( 10 ^ M ) )
31	30	oveq1i 6660	. . . . . . 7 |- ( ( ( A x. ( 10 ^ M ) ) x. C ) + S ) = ( ( R x. ( 10 ^ M ) ) + S )
32		karatsubaOLD.w	. . . . . . 7 |- ( ( R x. ( 10 ^ M ) ) + S ) = W
33	31, 32	eqtri 2644	. . . . . 6 |- ( ( ( A x. ( 10 ^ M ) ) x. C ) + S ) = W
34	33	oveq1i 6660	. . . . 5 |- ( ( ( ( A x. ( 10 ^ M ) ) x. C ) + S ) x. ( 10 ^ M ) ) = ( W x. ( 10 ^ M ) )
35	8, 12, 7	mulassi 10049	. . . . . 6 |- ( ( ( A x. ( 10 ^ M ) ) x. C ) x. ( 10 ^ M ) ) = ( ( A x. ( 10 ^ M ) ) x. ( C x. ( 10 ^ M ) ) )
36	2, 12	mulcli 10045	. . . . . . . . . . . 12 |- ( A x. C ) e. CC
37	36, 18, 25	add32i 10259	. . . . . . . . . . 11 |- ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( ( A x. C ) + S ) + ( D x. B ) )
38	28	oveq1i 6660	. . . . . . . . . . . 12 |- ( ( A x. C ) + S ) = ( R + S )
39		karatsubaOLD.t	. . . . . . . . . . . . 13 |- ( B x. D ) = T
40	10, 15, 39	mulcomli 10047	. . . . . . . . . . . 12 |- ( D x. B ) = T
41	38, 40	oveq12i 6662	. . . . . . . . . . 11 |- ( ( ( A x. C ) + S ) + ( D x. B ) ) = ( ( R + S ) + T )
42	37, 41	eqtri 2644	. . . . . . . . . 10 |- ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( R + S ) + T )
43		karatsubaOLD.e	. . . . . . . . . 10 |- ( ( A + B ) x. ( C + D ) ) = ( ( R + S ) + T )
44	2, 10, 12, 15	muladdi 10481	. . . . . . . . . 10 |- ( ( A + B ) x. ( C + D ) ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )
45	42, 43, 44	3eqtr2i 2650	. . . . . . . . 9 |- ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )
46	36, 18	addcli 10044	. . . . . . . . . 10 |- ( ( A x. C ) + ( D x. B ) ) e. CC
47	2, 15	mulcli 10045	. . . . . . . . . . 11 |- ( A x. D ) e. CC
48	12, 10	mulcli 10045	. . . . . . . . . . 11 |- ( C x. B ) e. CC
49	47, 48	addcli 10044	. . . . . . . . . 10 |- ( ( A x. D ) + ( C x. B ) ) e. CC
50	46, 25, 49	addcani 10229	. . . . . . . . 9 |- ( ( ( ( A x. C ) + ( D x. B ) ) + S ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) ) <-> S = ( ( A x. D ) + ( C x. B ) ) )
51	45, 50	mpbi 220	. . . . . . . 8 |- S = ( ( A x. D ) + ( C x. B ) )
52	51	oveq1i 6660	. . . . . . 7 |- ( S x. ( 10 ^ M ) ) = ( ( ( A x. D ) + ( C x. B ) ) x. ( 10 ^ M ) )
53	47, 48, 7	adddiri 10051	. . . . . . 7 |- ( ( ( A x. D ) + ( C x. B ) ) x. ( 10 ^ M ) ) = ( ( ( A x. D ) x. ( 10 ^ M ) ) + ( ( C x. B ) x. ( 10 ^ M ) ) )
54	2, 15, 7	mul32i 10232	. . . . . . . 8 |- ( ( A x. D ) x. ( 10 ^ M ) ) = ( ( A x. ( 10 ^ M ) ) x. D )
55	12, 10, 7	mul32i 10232	. . . . . . . 8 |- ( ( C x. B ) x. ( 10 ^ M ) ) = ( ( C x. ( 10 ^ M ) ) x. B )
56	54, 55	oveq12i 6662	. . . . . . 7 |- ( ( ( A x. D ) x. ( 10 ^ M ) ) + ( ( C x. B ) x. ( 10 ^ M ) ) ) = ( ( ( A x. ( 10 ^ M ) ) x. D ) + ( ( C x. ( 10 ^ M ) ) x. B ) )
57	52, 53, 56	3eqtri 2648	. . . . . 6 |- ( S x. ( 10 ^ M ) ) = ( ( ( A x. ( 10 ^ M ) ) x. D ) + ( ( C x. ( 10 ^ M ) ) x. B ) )
58	35, 57	oveq12i 6662	. . . . 5 |- ( ( ( ( A x. ( 10 ^ M ) ) x. C ) x. ( 10 ^ M ) ) + ( S x. ( 10 ^ M ) ) ) = ( ( ( A x. ( 10 ^ M ) ) x. ( C x. ( 10 ^ M ) ) ) + ( ( ( A x. ( 10 ^ M ) ) x. D ) + ( ( C x. ( 10 ^ M ) ) x. B ) ) )
59	26, 34, 58	3eqtr3ri 2653	. . . 4 |- ( ( ( A x. ( 10 ^ M ) ) x. ( C x. ( 10 ^ M ) ) ) + ( ( ( A x. ( 10 ^ M ) ) x. D ) + ( ( C x. ( 10 ^ M ) ) x. B ) ) ) = ( W x. ( 10 ^ M ) )
60	59, 40	oveq12i 6662	. . 3 |- ( ( ( ( A x. ( 10 ^ M ) ) x. ( C x. ( 10 ^ M ) ) ) + ( ( ( A x. ( 10 ^ M ) ) x. D ) + ( ( C x. ( 10 ^ M ) ) x. B ) ) ) + ( D x. B ) ) = ( ( W x. ( 10 ^ M ) ) + T )
61	16, 22, 60	3eqtri 2648	. 2 |- ( ( ( A x. ( 10 ^ M ) ) + B ) x. ( ( C x. ( 10 ^ M ) ) + D ) ) = ( ( W x. ( 10 ^ M ) ) + T )
62		karatsubaOLD.x	. . 3 |- ( ( A x. ( 10 ^ M ) ) + B ) = X
63		karatsubaOLD.y	. . 3 |- ( ( C x. ( 10 ^ M ) ) + D ) = Y
64	62, 63	oveq12i 6662	. 2 |- ( ( ( A x. ( 10 ^ M ) ) + B ) x. ( ( C x. ( 10 ^ M ) ) + D ) ) = ( X x. Y )
65		karatsubaOLD.z	. 2 |- ( ( W x. ( 10 ^ M ) ) + T ) = Z
66	61, 64, 65	3eqtr3i 2652	1 |- ( X x. Y ) = Z

Syntax hints:   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   + caddc 9939   x. cmul 9941   10c10 11078   NN0cn0 11292   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-10OLD 11087  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861

This theorem is referenced by: (None)