Theorem dpmul 29621

Description: Multiplication with one decimal point. (Contributed by Thierry Arnoux, 26-Dec-2021.)

Hypotheses

Ref	Expression
dpmul.a	|- A e. NN0
dpmul.b	|- B e. NN0
dpmul.c	|- C e. NN0
dpmul.d	|- D e. NN0
dpmul.e	|- E e. NN0
dpmul.g	|- G e. NN0
dpmul.j	|- J e. NN0
dpmul.k	|- K e. NN0
dpmul.1	|- ( A x. C ) = F
dpmul.2	|- ( A x. D ) = M
dpmul.3	|- ( B x. C ) = L
dpmul.4	|- ( B x. D ) = ; E K
dpmul.5	|- ( ( L + M ) + E ) = ; G J
dpmul.6	|- ( F + G ) = I

Assertion

Ref	Expression
dpmul	|- ( ( A period B ) x. ( C period D ) ) = ( I period_ J K )

Proof of Theorem dpmul

Step	Hyp	Ref	Expression
1		dpmul.a	. . . . 5 |- A e. NN0
2		dpmul.b	. . . . 5 |- B e. NN0
3	1, 2	deccl 11512	. . . 4 |- ; A B e. NN0
4		dpmul.c	. . . 4 |- C e. NN0
5		dpmul.d	. . . 4 |- D e. NN0
6		eqid 2622	. . . 4 |- ; C D = ; C D
7		dpmul.k	. . . 4 |- K e. NN0
8		dpmul.2	. . . . . 6 |- ( A x. D ) = M
9	1, 5	nn0mulcli 11331	. . . . . 6 |- ( A x. D ) e. NN0
10	8, 9	eqeltrri 2698	. . . . 5 |- M e. NN0
11		dpmul.e	. . . . 5 |- E e. NN0
12	10, 11	nn0addcli 11330	. . . 4 |- ( M + E ) e. NN0
13		eqid 2622	. . . . . . 7 |- ; A B = ; A B
14		dpmul.3	. . . . . . . 8 |- ( B x. C ) = L
15	2, 4	nn0mulcli 11331	. . . . . . . 8 |- ( B x. C ) e. NN0
16	14, 15	eqeltrri 2698	. . . . . . 7 |- L e. NN0
17		dpmul.1	. . . . . . 7 |- ( A x. C ) = F
18	4, 1, 2, 13, 16, 17, 14	decmul1 11585	. . . . . 6 |- (; A B x. C ) = ; F L
19	18	oveq1i 6660	. . . . 5 |- ( (; A B x. C ) + ( M + E ) ) = (; F L + ( M + E ) )
20		dfdec10 11497	. . . . . 6 |- ; F L = ( (; 1 0 x. F ) + L )
21	20	oveq1i 6660	. . . . 5 |- (; F L + ( M + E ) ) = ( ( (; 1 0 x. F ) + L ) + ( M + E ) )
22		10nn0 11516	. . . . . . . . 9 |- ; 1 0 e. NN0
23	22	nn0cni 11304	. . . . . . . 8 |- ; 1 0 e. CC
24	1, 4	nn0mulcli 11331	. . . . . . . . . 10 |- ( A x. C ) e. NN0
25	17, 24	eqeltrri 2698	. . . . . . . . 9 |- F e. NN0
26	25	nn0cni 11304	. . . . . . . 8 |- F e. CC
27	23, 26	mulcli 10045	. . . . . . 7 |- (; 1 0 x. F ) e. CC
28	16	nn0cni 11304	. . . . . . 7 |- L e. CC
29	12	nn0cni 11304	. . . . . . 7 |- ( M + E ) e. CC
30	27, 28, 29	addassi 10048	. . . . . 6 |- ( ( (; 1 0 x. F ) + L ) + ( M + E ) ) = ( (; 1 0 x. F ) + ( L + ( M + E ) ) )
31		dpmul.5	. . . . . . . 8 |- ( ( L + M ) + E ) = ; G J
32	10	nn0cni 11304	. . . . . . . . 9 |- M e. CC
33	11	nn0cni 11304	. . . . . . . . 9 |- E e. CC
34	28, 32, 33	addassi 10048	. . . . . . . 8 |- ( ( L + M ) + E ) = ( L + ( M + E ) )
35		dfdec10 11497	. . . . . . . 8 |- ; G J = ( (; 1 0 x. G ) + J )
36	31, 34, 35	3eqtr3ri 2653	. . . . . . 7 |- ( (; 1 0 x. G ) + J ) = ( L + ( M + E ) )
37	36	oveq2i 6661	. . . . . 6 |- ( (; 1 0 x. F ) + ( (; 1 0 x. G ) + J ) ) = ( (; 1 0 x. F ) + ( L + ( M + E ) ) )
38		dfdec10 11497	. . . . . . 7 |- ; I J = ( (; 1 0 x. I ) + J )
39		dpmul.g	. . . . . . . . . . 11 |- G e. NN0
40	39	nn0cni 11304	. . . . . . . . . 10 |- G e. CC
41	23, 26, 40	adddii 10050	. . . . . . . . 9 |- (; 1 0 x. ( F + G ) ) = ( (; 1 0 x. F ) + (; 1 0 x. G ) )
42		dpmul.6	. . . . . . . . . 10 |- ( F + G ) = I
43	42	oveq2i 6661	. . . . . . . . 9 |- (; 1 0 x. ( F + G ) ) = (; 1 0 x. I )
44	41, 43	eqtr3i 2646	. . . . . . . 8 |- ( (; 1 0 x. F ) + (; 1 0 x. G ) ) = (; 1 0 x. I )
45	44	oveq1i 6660	. . . . . . 7 |- ( ( (; 1 0 x. F ) + (; 1 0 x. G ) ) + J ) = ( (; 1 0 x. I ) + J )
46	23, 40	mulcli 10045	. . . . . . . 8 |- (; 1 0 x. G ) e. CC
47		dpmul.j	. . . . . . . . 9 |- J e. NN0
48	47	nn0cni 11304	. . . . . . . 8 |- J e. CC
49	27, 46, 48	addassi 10048	. . . . . . 7 |- ( ( (; 1 0 x. F ) + (; 1 0 x. G ) ) + J ) = ( (; 1 0 x. F ) + ( (; 1 0 x. G ) + J ) )
50	38, 45, 49	3eqtr2ri 2651	. . . . . 6 |- ( (; 1 0 x. F ) + ( (; 1 0 x. G ) + J ) ) = ; I J
51	30, 37, 50	3eqtr2i 2650	. . . . 5 |- ( ( (; 1 0 x. F ) + L ) + ( M + E ) ) = ; I J
52	19, 21, 51	3eqtri 2648	. . . 4 |- ( (; A B x. C ) + ( M + E ) ) = ; I J
53	8	oveq1i 6660	. . . . 5 |- ( ( A x. D ) + E ) = ( M + E )
54		dpmul.4	. . . . 5 |- ( B x. D ) = ; E K
55	5, 1, 2, 13, 7, 11, 53, 54	decmul1c 11587	. . . 4 |- (; A B x. D ) = ; ( M + E ) K
56	3, 4, 5, 6, 7, 12, 52, 55	decmul2c 11589	. . 3 |- (; A B x. ; C D ) = ;; I J K
57	2	nn0rei 11303	. . . . . . 7 |- B e. RR
58		dpcl 29598	. . . . . . 7 |- ( ( A e. NN0 /\ B e. RR ) -> ( A period B ) e. RR )
59	1, 57, 58	mp2an 708	. . . . . 6 |- ( A period B ) e. RR
60	59	recni 10052	. . . . 5 |- ( A period B ) e. CC
61	5	nn0rei 11303	. . . . . . 7 |- D e. RR
62		dpcl 29598	. . . . . . 7 |- ( ( C e. NN0 /\ D e. RR ) -> ( C period D ) e. RR )
63	4, 61, 62	mp2an 708	. . . . . 6 |- ( C period D ) e. RR
64	63	recni 10052	. . . . 5 |- ( C period D ) e. CC
65	60, 64, 23, 23	mul4i 10233	. . . 4 |- ( ( ( A period B ) x. ( C period D ) ) x. (; 1 0 x. ; 1 0 ) ) = ( ( ( A period B ) x. ; 1 0 ) x. ( ( C period D ) x. ; 1 0 ) )
66	22	dec0u 11520	. . . . 5 |- (; 1 0 x. ; 1 0 ) = ;; 1 0 0
67	66	oveq2i 6661	. . . 4 |- ( ( ( A period B ) x. ( C period D ) ) x. (; 1 0 x. ; 1 0 ) ) = ( ( ( A period B ) x. ( C period D ) ) x. ;; 1 0 0 )
68	1, 57	dpmul10 29603	. . . . 5 |- ( ( A period B ) x. ; 1 0 ) = ; A B
69	4, 61	dpmul10 29603	. . . . 5 |- ( ( C period D ) x. ; 1 0 ) = ; C D
70	68, 69	oveq12i 6662	. . . 4 |- ( ( ( A period B ) x. ; 1 0 ) x. ( ( C period D ) x. ; 1 0 ) ) = (; A B x. ; C D )
71	65, 67, 70	3eqtr3i 2652	. . 3 |- ( ( ( A period B ) x. ( C period D ) ) x. ;; 1 0 0 ) = (; A B x. ; C D )
72	25, 39	nn0addcli 11330	. . . . 5 |- ( F + G ) e. NN0
73	42, 72	eqeltrri 2698	. . . 4 |- I e. NN0
74	7	nn0rei 11303	. . . 4 |- K e. RR
75	73, 47, 74	dpmul100 29605	. . 3 |- ( ( I period_ J K ) x. ;; 1 0 0 ) = ;; I J K
76	56, 71, 75	3eqtr4i 2654	. 2 |- ( ( ( A period B ) x. ( C period D ) ) x. ;; 1 0 0 ) = ( ( I period_ J K ) x. ;; 1 0 0 )
77	60, 64	mulcli 10045	. . 3 |- ( ( A period B ) x. ( C period D ) ) e. CC
78	47	nn0rei 11303	. . . . . 6 |- J e. RR
79		dp2cl 29587	. . . . . 6 |- ( ( J e. RR /\ K e. RR ) -> _ J K e. RR )
80	78, 74, 79	mp2an 708	. . . . 5 |- _ J K e. RR
81		dpcl 29598	. . . . 5 |- ( ( I e. NN0 /\ _ J K e. RR ) -> ( I period_ J K ) e. RR )
82	73, 80, 81	mp2an 708	. . . 4 |- ( I period_ J K ) e. RR
83	82	recni 10052	. . 3 |- ( I period_ J K ) e. CC
84		10nn 11514	. . . . . 6 |- ; 1 0 e. NN
85	84	decnncl2 11525	. . . . 5 |- ;; 1 0 0 e. NN
86	85	nncni 11030	. . . 4 |- ;; 1 0 0 e. CC
87	85	nnne0i 11055	. . . 4 |- ;; 1 0 0 =/= 0
88	86, 87	pm3.2i 471	. . 3 |- (;; 1 0 0 e. CC /\ ;; 1 0 0 =/= 0 )
89		mulcan2 10665	. . 3 |- ( ( ( ( A period B ) x. ( C period D ) ) e. CC /\ ( I period_ J K ) e. CC /\ (;; 1 0 0 e. CC /\ ;; 1 0 0 =/= 0 ) ) -> ( ( ( ( A period B ) x. ( C period D ) ) x. ;; 1 0 0 ) = ( ( I period_ J K ) x. ;; 1 0 0 ) <-> ( ( A period B ) x. ( C period D ) ) = ( I period_ J K ) ) )
90	77, 83, 88, 89	mp3an 1424	. 2 |- ( ( ( ( A period B ) x. ( C period D ) ) x. ;; 1 0 0 ) = ( ( I period_ J K ) x. ;; 1 0 0 ) <-> ( ( A period B ) x. ( C period D ) ) = ( I period_ J K ) )
91	76, 90	mpbi 220	1 |- ( ( A period B ) x. ( C period D ) ) = ( I period_ J K )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NN0cn0 11292  ;cdc 11493  _cdp2 29577   periodcdp 29595

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-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-rmo 2920  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-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-div 10685  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-n0 11293  df-dec 11494  df-dp2 29578  df-dp 29596

This theorem is referenced by:  hgt750lem2  30730