dpmul - Mathbox for Thierry Arnoux

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Theorem dpmul 30615

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 12101	. . . 4 ⊢ ;𝐴𝐵 ∈ ℕ₀
4		dpmul.c	. . . 4 ⊢ 𝐶 ∈ ℕ₀
5		dpmul.d	. . . 4 ⊢ 𝐷 ∈ ℕ₀
6		eqid 2798	. . . 4 ⊢ ;𝐶𝐷 = ;𝐶𝐷
7		dpmul.k	. . . 4 ⊢ 𝐾 ∈ ℕ₀
8		dpmul.2	. . . . . 6 ⊢ (𝐴 · 𝐷) = 𝑀
9	1, 5	nn0mulcli 11923	. . . . . 6 ⊢ (𝐴 · 𝐷) ∈ ℕ₀
10	8, 9	eqeltrri 2887	. . . . 5 ⊢ 𝑀 ∈ ℕ₀
11		dpmul.e	. . . . 5 ⊢ 𝐸 ∈ ℕ₀
12	10, 11	nn0addcli 11922	. . . 4 ⊢ (𝑀 + 𝐸) ∈ ℕ₀
13		eqid 2798	. . . . . . 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 12150	. . . . . 6 ⊢ (;𝐴𝐵 · 𝐶) = ;𝐹𝐿
17	16	oveq1i 7145	. . . . 5 ⊢ ((;𝐴𝐵 · 𝐶) + (𝑀 + 𝐸)) = (;𝐹𝐿 + (𝑀 + 𝐸))
18		dfdec10 12089	. . . . . 6 ⊢ ;𝐹𝐿 = ((;10 · 𝐹) + 𝐿)
19	18	oveq1i 7145	. . . . 5 ⊢ (;𝐹𝐿 + (𝑀 + 𝐸)) = (((;10 · 𝐹) + 𝐿) + (𝑀 + 𝐸))
20		10nn0 12104	. . . . . . . . 9 ⊢ ;10 ∈ ℕ₀
21	20	nn0cni 11897	. . . . . . . 8 ⊢ ;10 ∈ ℂ
22	1, 4	nn0mulcli 11923	. . . . . . . . . 10 ⊢ (𝐴 · 𝐶) ∈ ℕ₀
23	14, 22	eqeltrri 2887	. . . . . . . . 9 ⊢ 𝐹 ∈ ℕ₀
24	23	nn0cni 11897	. . . . . . . 8 ⊢ 𝐹 ∈ ℂ
25	21, 24	mulcli 10637	. . . . . . 7 ⊢ (;10 · 𝐹) ∈ ℂ
26	2, 4	nn0mulcli 11923	. . . . . . . . 9 ⊢ (𝐵 · 𝐶) ∈ ℕ₀
27	15, 26	eqeltrri 2887	. . . . . . . 8 ⊢ 𝐿 ∈ ℕ₀
28	27	nn0cni 11897	. . . . . . 7 ⊢ 𝐿 ∈ ℂ
29	12	nn0cni 11897	. . . . . . 7 ⊢ (𝑀 + 𝐸) ∈ ℂ
30	25, 28, 29	addassi 10640	. . . . . 6 ⊢ (((;10 · 𝐹) + 𝐿) + (𝑀 + 𝐸)) = ((;10 · 𝐹) + (𝐿 + (𝑀 + 𝐸)))
31		dpmul.5	. . . . . . . 8 ⊢ ((𝐿 + 𝑀) + 𝐸) = ;𝐺𝐽
32	10	nn0cni 11897	. . . . . . . . 9 ⊢ 𝑀 ∈ ℂ
33	11	nn0cni 11897	. . . . . . . . 9 ⊢ 𝐸 ∈ ℂ
34	28, 32, 33	addassi 10640	. . . . . . . 8 ⊢ ((𝐿 + 𝑀) + 𝐸) = (𝐿 + (𝑀 + 𝐸))
35		dfdec10 12089	. . . . . . . 8 ⊢ ;𝐺𝐽 = ((;10 · 𝐺) + 𝐽)
36	31, 34, 35	3eqtr3ri 2830	. . . . . . 7 ⊢ ((;10 · 𝐺) + 𝐽) = (𝐿 + (𝑀 + 𝐸))
37	36	oveq2i 7146	. . . . . 6 ⊢ ((;10 · 𝐹) + ((;10 · 𝐺) + 𝐽)) = ((;10 · 𝐹) + (𝐿 + (𝑀 + 𝐸)))
38		dfdec10 12089	. . . . . . 7 ⊢ ;𝐼𝐽 = ((;10 · 𝐼) + 𝐽)
39		dpmul.g	. . . . . . . . . . 11 ⊢ 𝐺 ∈ ℕ₀
40	39	nn0cni 11897	. . . . . . . . . 10 ⊢ 𝐺 ∈ ℂ
41	21, 24, 40	adddii 10642	. . . . . . . . 9 ⊢ (;10 · (𝐹 + 𝐺)) = ((;10 · 𝐹) + (;10 · 𝐺))
42		dpmul.6	. . . . . . . . . 10 ⊢ (𝐹 + 𝐺) = 𝐼
43	42	oveq2i 7146	. . . . . . . . 9 ⊢ (;10 · (𝐹 + 𝐺)) = (;10 · 𝐼)
44	41, 43	eqtr3i 2823	. . . . . . . 8 ⊢ ((;10 · 𝐹) + (;10 · 𝐺)) = (;10 · 𝐼)
45	44	oveq1i 7145	. . . . . . 7 ⊢ (((;10 · 𝐹) + (;10 · 𝐺)) + 𝐽) = ((;10 · 𝐼) + 𝐽)
46	21, 40	mulcli 10637	. . . . . . . 8 ⊢ (;10 · 𝐺) ∈ ℂ
47		dpmul.j	. . . . . . . . 9 ⊢ 𝐽 ∈ ℕ₀
48	47	nn0cni 11897	. . . . . . . 8 ⊢ 𝐽 ∈ ℂ
49	25, 46, 48	addassi 10640	. . . . . . 7 ⊢ (((;10 · 𝐹) + (;10 · 𝐺)) + 𝐽) = ((;10 · 𝐹) + ((;10 · 𝐺) + 𝐽))
50	38, 45, 49	3eqtr2ri 2828	. . . . . 6 ⊢ ((;10 · 𝐹) + ((;10 · 𝐺) + 𝐽)) = ;𝐼𝐽
51	30, 37, 50	3eqtr2i 2827	. . . . 5 ⊢ (((;10 · 𝐹) + 𝐿) + (𝑀 + 𝐸)) = ;𝐼𝐽
52	17, 19, 51	3eqtri 2825	. . . 4 ⊢ ((;𝐴𝐵 · 𝐶) + (𝑀 + 𝐸)) = ;𝐼𝐽
53	8	oveq1i 7145	. . . . 5 ⊢ ((𝐴 · 𝐷) + 𝐸) = (𝑀 + 𝐸)
54		dpmul.4	. . . . 5 ⊢ (𝐵 · 𝐷) = ;𝐸𝐾
55	5, 1, 2, 13, 7, 11, 53, 54	decmul1c 12151	. . . 4 ⊢ (;𝐴𝐵 · 𝐷) = ;(𝑀 + 𝐸)𝐾
56	3, 4, 5, 6, 7, 12, 52, 55	decmul2c 12152	. . 3 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐼𝐽𝐾
57	2	nn0rei 11896	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
58		dpcl 30593	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
59	1, 57, 58	mp2an 691	. . . . . 6 ⊢ (𝐴.𝐵) ∈ ℝ
60	59	recni 10644	. . . . 5 ⊢ (𝐴.𝐵) ∈ ℂ
61	5	nn0rei 11896	. . . . . . 7 ⊢ 𝐷 ∈ ℝ
62		dpcl 30593	. . . . . . 7 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) ∈ ℝ)
63	4, 61, 62	mp2an 691	. . . . . 6 ⊢ (𝐶.𝐷) ∈ ℝ
64	63	recni 10644	. . . . 5 ⊢ (𝐶.𝐷) ∈ ℂ
65	60, 64, 21, 21	mul4i 10826	. . . 4 ⊢ (((𝐴.𝐵) · (𝐶.𝐷)) · (;10 · ;10)) = (((𝐴.𝐵) · ;10) · ((𝐶.𝐷) · ;10))
66	20	dec0u 12107	. . . . 5 ⊢ (;10 · ;10) = ;;100
67	66	oveq2i 7146	. . . 4 ⊢ (((𝐴.𝐵) · (𝐶.𝐷)) · (;10 · ;10)) = (((𝐴.𝐵) · (𝐶.𝐷)) · ;;100)
68	1, 57	dpmul10 30597	. . . . 5 ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵
69	4, 61	dpmul10 30597	. . . . 5 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷
70	68, 69	oveq12i 7147	. . . 4 ⊢ (((𝐴.𝐵) · ;10) · ((𝐶.𝐷) · ;10)) = (;𝐴𝐵 · ;𝐶𝐷)
71	65, 67, 70	3eqtr3i 2829	. . 3 ⊢ (((𝐴.𝐵) · (𝐶.𝐷)) · ;;100) = (;𝐴𝐵 · ;𝐶𝐷)
72	23, 39	nn0addcli 11922	. . . . 5 ⊢ (𝐹 + 𝐺) ∈ ℕ₀
73	42, 72	eqeltrri 2887	. . . 4 ⊢ 𝐼 ∈ ℕ₀
74	7	nn0rei 11896	. . . 4 ⊢ 𝐾 ∈ ℝ
75	73, 47, 74	dpmul100 30599	. . 3 ⊢ ((𝐼._𝐽𝐾) · ;;100) = ;;𝐼𝐽𝐾
76	56, 71, 75	3eqtr4i 2831	. 2 ⊢ (((𝐴.𝐵) · (𝐶.𝐷)) · ;;100) = ((𝐼._𝐽𝐾) · ;;100)
77	60, 64	mulcli 10637	. . 3 ⊢ ((𝐴.𝐵) · (𝐶.𝐷)) ∈ ℂ
78	47	nn0rei 11896	. . . . . 6 ⊢ 𝐽 ∈ ℝ
79		dp2cl 30582	. . . . . 6 ⊢ ((𝐽 ∈ ℝ ∧ 𝐾 ∈ ℝ) → _𝐽𝐾 ∈ ℝ)
80	78, 74, 79	mp2an 691	. . . . 5 ⊢ _𝐽𝐾 ∈ ℝ
81		dpcl 30593	. . . . 5 ⊢ ((𝐼 ∈ ℕ₀ ∧ _𝐽𝐾 ∈ ℝ) → (𝐼._𝐽𝐾) ∈ ℝ)
82	73, 80, 81	mp2an 691	. . . 4 ⊢ (𝐼._𝐽𝐾) ∈ ℝ
83	82	recni 10644	. . 3 ⊢ (𝐼._𝐽𝐾) ∈ ℂ
84		10nn 12102	. . . . . 6 ⊢ ;10 ∈ ℕ
85	84	decnncl2 12110	. . . . 5 ⊢ ;;100 ∈ ℕ
86	85	nncni 11635	. . . 4 ⊢ ;;100 ∈ ℂ
87	85	nnne0i 11665	. . . 4 ⊢ ;;100 ≠ 0
88	86, 87	pm3.2i 474	. . 3 ⊢ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)
89		mulcan2 11267	. . 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 1458	. 2 ⊢ ((((𝐴.𝐵) · (𝐶.𝐷)) · ;;100) = ((𝐼._𝐽𝐾) · ;;100) ↔ ((𝐴.𝐵) · (𝐶.𝐷)) = (𝐼._𝐽𝐾))
91	76, 90	mpbi 233	1 ⊢ ((𝐴.𝐵) · (𝐶.𝐷)) = (𝐼._𝐽𝐾)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ℕ₀cn0 11885 cdc 12086 cdp2 30573 .cdp 30590

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-dec 12087 df-dp2 30574 df-dp 30591

This theorem is referenced by: hgt750lem2 32033

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