dpmul - Mathbox for Thierry Arnoux

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Theorem dpmul 31187

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

**Hypotheses**
Ref	Expression
dpmul.a	⊢ 𝐴 ∈ ℕ₀
dpmul.b	⊢ 𝐵 ∈ ℕ₀
dpmul.c	⊢ 𝐶 ∈ ℕ₀
dpmul.d	⊢ 𝐷 ∈ ℕ₀
dpmul.e	⊢ 𝐸 ∈ ℕ₀
dpmul.g	⊢ 𝐺 ∈ ℕ₀
dpmul.j	⊢ 𝐽 ∈ ℕ₀
dpmul.k	⊢ 𝐾 ∈ ℕ₀
dpmul.1	⊢ (𝐴 · 𝐶) = 𝐹
dpmul.2	⊢ (𝐴 · 𝐷) = 𝑀
dpmul.3	⊢ (𝐵 · 𝐶) = 𝐿
dpmul.4	⊢ (𝐵 · 𝐷) = ;𝐸𝐾
dpmul.5	⊢ ((𝐿 + 𝑀) + 𝐸) = ;𝐺𝐽
dpmul.6	⊢ (𝐹 + 𝐺) = 𝐼

**Assertion**
Ref	Expression
dpmul	⊢ ((𝐴.𝐵) · (𝐶.𝐷)) = (𝐼._𝐽𝐾)

**Proof of Theorem dpmul**
Step	Hyp	Ref	Expression
1		dpmul.a	. . . . 5 ⊢ 𝐴 ∈ ℕ₀
2		dpmul.b	. . . . 5 ⊢ 𝐵 ∈ ℕ₀
3	1, 2	deccl 12452	. . . 4 ⊢ ;𝐴𝐵 ∈ ℕ₀
4		dpmul.c	. . . 4 ⊢ 𝐶 ∈ ℕ₀
5		dpmul.d	. . . 4 ⊢ 𝐷 ∈ ℕ₀
6		eqid 2738	. . . 4 ⊢ ;𝐶𝐷 = ;𝐶𝐷
7		dpmul.k	. . . 4 ⊢ 𝐾 ∈ ℕ₀
8		dpmul.2	. . . . . 6 ⊢ (𝐴 · 𝐷) = 𝑀
9	1, 5	nn0mulcli 12271	. . . . . 6 ⊢ (𝐴 · 𝐷) ∈ ℕ₀
10	8, 9	eqeltrri 2836	. . . . 5 ⊢ 𝑀 ∈ ℕ₀
11		dpmul.e	. . . . 5 ⊢ 𝐸 ∈ ℕ₀
12	10, 11	nn0addcli 12270	. . . 4 ⊢ (𝑀 + 𝐸) ∈ ℕ₀
13		eqid 2738	. . . . . . 7 ⊢ ;𝐴𝐵 = ;𝐴𝐵
14		dpmul.1	. . . . . . 7 ⊢ (𝐴 · 𝐶) = 𝐹
15		dpmul.3	. . . . . . 7 ⊢ (𝐵 · 𝐶) = 𝐿
16	4, 1, 2, 13, 14, 15	decmul1 12501	. . . . . 6 ⊢ (;𝐴𝐵 · 𝐶) = ;𝐹𝐿
17	16	oveq1i 7285	. . . . 5 ⊢ ((;𝐴𝐵 · 𝐶) + (𝑀 + 𝐸)) = (;𝐹𝐿 + (𝑀 + 𝐸))
18		dfdec10 12440	. . . . . 6 ⊢ ;𝐹𝐿 = ((;10 · 𝐹) + 𝐿)
19	18	oveq1i 7285	. . . . 5 ⊢ (;𝐹𝐿 + (𝑀 + 𝐸)) = (((;10 · 𝐹) + 𝐿) + (𝑀 + 𝐸))
20		10nn0 12455	. . . . . . . . 9 ⊢ ;10 ∈ ℕ₀
21	20	nn0cni 12245	. . . . . . . 8 ⊢ ;10 ∈ ℂ
22	1, 4	nn0mulcli 12271	. . . . . . . . . 10 ⊢ (𝐴 · 𝐶) ∈ ℕ₀
23	14, 22	eqeltrri 2836	. . . . . . . . 9 ⊢ 𝐹 ∈ ℕ₀
24	23	nn0cni 12245	. . . . . . . 8 ⊢ 𝐹 ∈ ℂ
25	21, 24	mulcli 10982	. . . . . . 7 ⊢ (;10 · 𝐹) ∈ ℂ
26	2, 4	nn0mulcli 12271	. . . . . . . . 9 ⊢ (𝐵 · 𝐶) ∈ ℕ₀
27	15, 26	eqeltrri 2836	. . . . . . . 8 ⊢ 𝐿 ∈ ℕ₀
28	27	nn0cni 12245	. . . . . . 7 ⊢ 𝐿 ∈ ℂ
29	12	nn0cni 12245	. . . . . . 7 ⊢ (𝑀 + 𝐸) ∈ ℂ
30	25, 28, 29	addassi 10985	. . . . . 6 ⊢ (((;10 · 𝐹) + 𝐿) + (𝑀 + 𝐸)) = ((;10 · 𝐹) + (𝐿 + (𝑀 + 𝐸)))
31		dpmul.5	. . . . . . . 8 ⊢ ((𝐿 + 𝑀) + 𝐸) = ;𝐺𝐽
32	10	nn0cni 12245	. . . . . . . . 9 ⊢ 𝑀 ∈ ℂ
33	11	nn0cni 12245	. . . . . . . . 9 ⊢ 𝐸 ∈ ℂ
34	28, 32, 33	addassi 10985	. . . . . . . 8 ⊢ ((𝐿 + 𝑀) + 𝐸) = (𝐿 + (𝑀 + 𝐸))
35		dfdec10 12440	. . . . . . . 8 ⊢ ;𝐺𝐽 = ((;10 · 𝐺) + 𝐽)
36	31, 34, 35	3eqtr3ri 2775	. . . . . . 7 ⊢ ((;10 · 𝐺) + 𝐽) = (𝐿 + (𝑀 + 𝐸))
37	36	oveq2i 7286	. . . . . 6 ⊢ ((;10 · 𝐹) + ((;10 · 𝐺) + 𝐽)) = ((;10 · 𝐹) + (𝐿 + (𝑀 + 𝐸)))
38		dfdec10 12440	. . . . . . 7 ⊢ ;𝐼𝐽 = ((;10 · 𝐼) + 𝐽)
39		dpmul.g	. . . . . . . . . . 11 ⊢ 𝐺 ∈ ℕ₀
40	39	nn0cni 12245	. . . . . . . . . 10 ⊢ 𝐺 ∈ ℂ
41	21, 24, 40	adddii 10987	. . . . . . . . 9 ⊢ (;10 · (𝐹 + 𝐺)) = ((;10 · 𝐹) + (;10 · 𝐺))
42		dpmul.6	. . . . . . . . . 10 ⊢ (𝐹 + 𝐺) = 𝐼
43	42	oveq2i 7286	. . . . . . . . 9 ⊢ (;10 · (𝐹 + 𝐺)) = (;10 · 𝐼)
44	41, 43	eqtr3i 2768	. . . . . . . 8 ⊢ ((;10 · 𝐹) + (;10 · 𝐺)) = (;10 · 𝐼)
45	44	oveq1i 7285	. . . . . . 7 ⊢ (((;10 · 𝐹) + (;10 · 𝐺)) + 𝐽) = ((;10 · 𝐼) + 𝐽)
46	21, 40	mulcli 10982	. . . . . . . 8 ⊢ (;10 · 𝐺) ∈ ℂ
47		dpmul.j	. . . . . . . . 9 ⊢ 𝐽 ∈ ℕ₀
48	47	nn0cni 12245	. . . . . . . 8 ⊢ 𝐽 ∈ ℂ
49	25, 46, 48	addassi 10985	. . . . . . 7 ⊢ (((;10 · 𝐹) + (;10 · 𝐺)) + 𝐽) = ((;10 · 𝐹) + ((;10 · 𝐺) + 𝐽))
50	38, 45, 49	3eqtr2ri 2773	. . . . . 6 ⊢ ((;10 · 𝐹) + ((;10 · 𝐺) + 𝐽)) = ;𝐼𝐽
51	30, 37, 50	3eqtr2i 2772	. . . . 5 ⊢ (((;10 · 𝐹) + 𝐿) + (𝑀 + 𝐸)) = ;𝐼𝐽
52	17, 19, 51	3eqtri 2770	. . . 4 ⊢ ((;𝐴𝐵 · 𝐶) + (𝑀 + 𝐸)) = ;𝐼𝐽
53	8	oveq1i 7285	. . . . 5 ⊢ ((𝐴 · 𝐷) + 𝐸) = (𝑀 + 𝐸)
54		dpmul.4	. . . . 5 ⊢ (𝐵 · 𝐷) = ;𝐸𝐾
55	5, 1, 2, 13, 7, 11, 53, 54	decmul1c 12502	. . . 4 ⊢ (;𝐴𝐵 · 𝐷) = ;(𝑀 + 𝐸)𝐾
56	3, 4, 5, 6, 7, 12, 52, 55	decmul2c 12503	. . 3 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐼𝐽𝐾
57	2	nn0rei 12244	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
58		dpcl 31165	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
59	1, 57, 58	mp2an 689	. . . . . 6 ⊢ (𝐴.𝐵) ∈ ℝ
60	59	recni 10989	. . . . 5 ⊢ (𝐴.𝐵) ∈ ℂ
61	5	nn0rei 12244	. . . . . . 7 ⊢ 𝐷 ∈ ℝ
62		dpcl 31165	. . . . . . 7 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) ∈ ℝ)
63	4, 61, 62	mp2an 689	. . . . . 6 ⊢ (𝐶.𝐷) ∈ ℝ
64	63	recni 10989	. . . . 5 ⊢ (𝐶.𝐷) ∈ ℂ
65	60, 64, 21, 21	mul4i 11172	. . . 4 ⊢ (((𝐴.𝐵) · (𝐶.𝐷)) · (;10 · ;10)) = (((𝐴.𝐵) · ;10) · ((𝐶.𝐷) · ;10))
66	20	dec0u 12458	. . . . 5 ⊢ (;10 · ;10) = ;;100
67	66	oveq2i 7286	. . . 4 ⊢ (((𝐴.𝐵) · (𝐶.𝐷)) · (;10 · ;10)) = (((𝐴.𝐵) · (𝐶.𝐷)) · ;;100)
68	1, 57	dpmul10 31169	. . . . 5 ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵
69	4, 61	dpmul10 31169	. . . . 5 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷
70	68, 69	oveq12i 7287	. . . 4 ⊢ (((𝐴.𝐵) · ;10) · ((𝐶.𝐷) · ;10)) = (;𝐴𝐵 · ;𝐶𝐷)
71	65, 67, 70	3eqtr3i 2774	. . 3 ⊢ (((𝐴.𝐵) · (𝐶.𝐷)) · ;;100) = (;𝐴𝐵 · ;𝐶𝐷)
72	23, 39	nn0addcli 12270	. . . . 5 ⊢ (𝐹 + 𝐺) ∈ ℕ₀
73	42, 72	eqeltrri 2836	. . . 4 ⊢ 𝐼 ∈ ℕ₀
74	7	nn0rei 12244	. . . 4 ⊢ 𝐾 ∈ ℝ
75	73, 47, 74	dpmul100 31171	. . 3 ⊢ ((𝐼._𝐽𝐾) · ;;100) = ;;𝐼𝐽𝐾
76	56, 71, 75	3eqtr4i 2776	. 2 ⊢ (((𝐴.𝐵) · (𝐶.𝐷)) · ;;100) = ((𝐼._𝐽𝐾) · ;;100)
77	60, 64	mulcli 10982	. . 3 ⊢ ((𝐴.𝐵) · (𝐶.𝐷)) ∈ ℂ
78	47	nn0rei 12244	. . . . . 6 ⊢ 𝐽 ∈ ℝ
79		dp2cl 31154	. . . . . 6 ⊢ ((𝐽 ∈ ℝ ∧ 𝐾 ∈ ℝ) → _𝐽𝐾 ∈ ℝ)
80	78, 74, 79	mp2an 689	. . . . 5 ⊢ _𝐽𝐾 ∈ ℝ
81		dpcl 31165	. . . . 5 ⊢ ((𝐼 ∈ ℕ₀ ∧ _𝐽𝐾 ∈ ℝ) → (𝐼._𝐽𝐾) ∈ ℝ)
82	73, 80, 81	mp2an 689	. . . 4 ⊢ (𝐼._𝐽𝐾) ∈ ℝ
83	82	recni 10989	. . 3 ⊢ (𝐼._𝐽𝐾) ∈ ℂ
84		10nn 12453	. . . . . 6 ⊢ ;10 ∈ ℕ
85	84	decnncl2 12461	. . . . 5 ⊢ ;;100 ∈ ℕ
86	85	nncni 11983	. . . 4 ⊢ ;;100 ∈ ℂ
87	85	nnne0i 12013	. . . 4 ⊢ ;;100 ≠ 0
88	86, 87	pm3.2i 471	. . 3 ⊢ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)
89		mulcan2 11613	. . 3 ⊢ ((((𝐴.𝐵) · (𝐶.𝐷)) ∈ ℂ ∧ (𝐼._𝐽𝐾) ∈ ℂ ∧ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)) → ((((𝐴.𝐵) · (𝐶.𝐷)) · ;;100) = ((𝐼._𝐽𝐾) · ;;100) ↔ ((𝐴.𝐵) · (𝐶.𝐷)) = (𝐼._𝐽𝐾)))
90	77, 83, 88, 89	mp3an 1460	. 2 ⊢ ((((𝐴.𝐵) · (𝐶.𝐷)) · ;;100) = ((𝐼._𝐽𝐾) · ;;100) ↔ ((𝐴.𝐵) · (𝐶.𝐷)) = (𝐼._𝐽𝐾))
91	76, 90	mpbi 229	1 ⊢ ((𝐴.𝐵) · (𝐶.𝐷)) = (𝐼._𝐽𝐾)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 ℕ₀cn0 12233 cdc 12437 cdp2 31145 .cdp 31162

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-dec 12438 df-dp2 31146 df-dp 31163

This theorem is referenced by: hgt750lem2 32632

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