dpmul - Mathbox for Thierry Arnoux

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Theorem dpmul 33549

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 12856	. . . 4 ⊢ ;𝐴𝐵 ∈ ℕ₀
4		dpmul.c	. . . 4 ⊢ 𝐶 ∈ ℕ₀
5		dpmul.d	. . . 4 ⊢ 𝐷 ∈ ℕ₀
6		eqid 2763	. . . 4 ⊢ ;𝐶𝐷 = ;𝐶𝐷
7		dpmul.k	. . . 4 ⊢ 𝐾 ∈ ℕ₀
8		dpmul.2	. . . . . 6 ⊢ (𝐴 · 𝐷) = 𝑀
9	1, 5	nn0mulcli 12671	. . . . . 6 ⊢ (𝐴 · 𝐷) ∈ ℕ₀
10	8, 9	eqeltrri 2861	. . . . 5 ⊢ 𝑀 ∈ ℕ₀
11		dpmul.e	. . . . 5 ⊢ 𝐸 ∈ ℕ₀
12	10, 11	nn0addcli 12670	. . . 4 ⊢ (𝑀 + 𝐸) ∈ ℕ₀
13		eqid 2763	. . . . . . 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 12910	. . . . . 6 ⊢ (;𝐴𝐵 · 𝐶) = ;𝐹𝐿
17	16	oveq1i 7453	. . . . 5 ⊢ ((;𝐴𝐵 · 𝐶) + (𝑀 + 𝐸)) = (;𝐹𝐿 + (𝑀 + 𝐸))
18		dfdec10 12844	. . . . . 6 ⊢ ;𝐹𝐿 = ((;10 · 𝐹) + 𝐿)
19	18	oveq1i 7453	. . . . 5 ⊢ (;𝐹𝐿 + (𝑀 + 𝐸)) = (((;10 · 𝐹) + 𝐿) + (𝑀 + 𝐸))
20		10nn0 12863	. . . . . . . . 9 ⊢ ;10 ∈ ℕ₀
21	20	nn0cni 12645	. . . . . . . 8 ⊢ ;10 ∈ ℂ
22	1, 4	nn0mulcli 12671	. . . . . . . . . 10 ⊢ (𝐴 · 𝐶) ∈ ℕ₀
23	14, 22	eqeltrri 2861	. . . . . . . . 9 ⊢ 𝐹 ∈ ℕ₀
24	23	nn0cni 12645	. . . . . . . 8 ⊢ 𝐹 ∈ ℂ
25	21, 24	mulcli 11343	. . . . . . 7 ⊢ (;10 · 𝐹) ∈ ℂ
26	2, 4	nn0mulcli 12671	. . . . . . . . 9 ⊢ (𝐵 · 𝐶) ∈ ℕ₀
27	15, 26	eqeltrri 2861	. . . . . . . 8 ⊢ 𝐿 ∈ ℕ₀
28	27	nn0cni 12645	. . . . . . 7 ⊢ 𝐿 ∈ ℂ
29	12	nn0cni 12645	. . . . . . 7 ⊢ (𝑀 + 𝐸) ∈ ℂ
30	25, 28, 29	addassi 11346	. . . . . 6 ⊢ (((;10 · 𝐹) + 𝐿) + (𝑀 + 𝐸)) = ((;10 · 𝐹) + (𝐿 + (𝑀 + 𝐸)))
31		dpmul.5	. . . . . . . 8 ⊢ ((𝐿 + 𝑀) + 𝐸) = ;𝐺𝐽
32	10	nn0cni 12645	. . . . . . . . 9 ⊢ 𝑀 ∈ ℂ
33	11	nn0cni 12645	. . . . . . . . 9 ⊢ 𝐸 ∈ ℂ
34	28, 32, 33	addassi 11346	. . . . . . . 8 ⊢ ((𝐿 + 𝑀) + 𝐸) = (𝐿 + (𝑀 + 𝐸))
35		dfdec10 12844	. . . . . . . 8 ⊢ ;𝐺𝐽 = ((;10 · 𝐺) + 𝐽)
36	31, 34, 35	3eqtr3ri 2795	. . . . . . 7 ⊢ ((;10 · 𝐺) + 𝐽) = (𝐿 + (𝑀 + 𝐸))
37	36	oveq2i 7454	. . . . . 6 ⊢ ((;10 · 𝐹) + ((;10 · 𝐺) + 𝐽)) = ((;10 · 𝐹) + (𝐿 + (𝑀 + 𝐸)))
38		dfdec10 12844	. . . . . . 7 ⊢ ;𝐼𝐽 = ((;10 · 𝐼) + 𝐽)
39		dpmul.g	. . . . . . . . . . 11 ⊢ 𝐺 ∈ ℕ₀
40	39	nn0cni 12645	. . . . . . . . . 10 ⊢ 𝐺 ∈ ℂ
41	21, 24, 40	adddii 11348	. . . . . . . . 9 ⊢ (;10 · (𝐹 + 𝐺)) = ((;10 · 𝐹) + (;10 · 𝐺))
42		dpmul.6	. . . . . . . . . 10 ⊢ (𝐹 + 𝐺) = 𝐼
43	42	oveq2i 7454	. . . . . . . . 9 ⊢ (;10 · (𝐹 + 𝐺)) = (;10 · 𝐼)
44	41, 43	eqtr3i 2788	. . . . . . . 8 ⊢ ((;10 · 𝐹) + (;10 · 𝐺)) = (;10 · 𝐼)
45	44	oveq1i 7453	. . . . . . 7 ⊢ (((;10 · 𝐹) + (;10 · 𝐺)) + 𝐽) = ((;10 · 𝐼) + 𝐽)
46	21, 40	mulcli 11343	. . . . . . . 8 ⊢ (;10 · 𝐺) ∈ ℂ
47		dpmul.j	. . . . . . . . 9 ⊢ 𝐽 ∈ ℕ₀
48	47	nn0cni 12645	. . . . . . . 8 ⊢ 𝐽 ∈ ℂ
49	25, 46, 48	addassi 11346	. . . . . . 7 ⊢ (((;10 · 𝐹) + (;10 · 𝐺)) + 𝐽) = ((;10 · 𝐹) + ((;10 · 𝐺) + 𝐽))
50	38, 45, 49	3eqtr2ri 2793	. . . . . 6 ⊢ ((;10 · 𝐹) + ((;10 · 𝐺) + 𝐽)) = ;𝐼𝐽
51	30, 37, 50	3eqtr2i 2792	. . . . 5 ⊢ (((;10 · 𝐹) + 𝐿) + (𝑀 + 𝐸)) = ;𝐼𝐽
52	17, 19, 51	3eqtri 2790	. . . 4 ⊢ ((;𝐴𝐵 · 𝐶) + (𝑀 + 𝐸)) = ;𝐼𝐽
53	8	oveq1i 7453	. . . . 5 ⊢ ((𝐴 · 𝐷) + 𝐸) = (𝑀 + 𝐸)
54		dpmul.4	. . . . 5 ⊢ (𝐵 · 𝐷) = ;𝐸𝐾
55	5, 1, 2, 13, 7, 11, 53, 54	decmul1c 12911	. . . 4 ⊢ (;𝐴𝐵 · 𝐷) = ;(𝑀 + 𝐸)𝐾
56	3, 4, 5, 6, 7, 12, 52, 55	decmul2c 12912	. . 3 ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐼𝐽𝐾
57	2	nn0rei 12644	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
58		dpcl 33527	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) ∈ ℝ)
59	1, 57, 58	mp2an 705	. . . . . 6 ⊢ (𝐴.𝐵) ∈ ℝ
60	59	recni 11350	. . . . 5 ⊢ (𝐴.𝐵) ∈ ℂ
61	5	nn0rei 12644	. . . . . . 7 ⊢ 𝐷 ∈ ℝ
62		dpcl 33527	. . . . . . 7 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) ∈ ℝ)
63	4, 61, 62	mp2an 705	. . . . . 6 ⊢ (𝐶.𝐷) ∈ ℝ
64	63	recni 11350	. . . . 5 ⊢ (𝐶.𝐷) ∈ ℂ
65	60, 64, 21, 21	mul4i 11534	. . . 4 ⊢ (((𝐴.𝐵) · (𝐶.𝐷)) · (;10 · ;10)) = (((𝐴.𝐵) · ;10) · ((𝐶.𝐷) · ;10))
66	20	dec0u 12867	. . . . 5 ⊢ (;10 · ;10) = ;;100
67	66	oveq2i 7454	. . . 4 ⊢ (((𝐴.𝐵) · (𝐶.𝐷)) · (;10 · ;10)) = (((𝐴.𝐵) · (𝐶.𝐷)) · ;;100)
68	1, 57	dpmul10 33531	. . . . 5 ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵
69	4, 61	dpmul10 33531	. . . . 5 ⊢ ((𝐶.𝐷) · ;10) = ;𝐶𝐷
70	68, 69	oveq12i 7455	. . . 4 ⊢ (((𝐴.𝐵) · ;10) · ((𝐶.𝐷) · ;10)) = (;𝐴𝐵 · ;𝐶𝐷)
71	65, 67, 70	3eqtr3i 2794	. . 3 ⊢ (((𝐴.𝐵) · (𝐶.𝐷)) · ;;100) = (;𝐴𝐵 · ;𝐶𝐷)
72	23, 39	nn0addcli 12670	. . . . 5 ⊢ (𝐹 + 𝐺) ∈ ℕ₀
73	42, 72	eqeltrri 2861	. . . 4 ⊢ 𝐼 ∈ ℕ₀
74	7	nn0rei 12644	. . . 4 ⊢ 𝐾 ∈ ℝ
75	73, 47, 74	dpmul100 33533	. . 3 ⊢ ((𝐼._𝐽𝐾) · ;;100) = ;;𝐼𝐽𝐾
76	56, 71, 75	3eqtr4i 2796	. 2 ⊢ (((𝐴.𝐵) · (𝐶.𝐷)) · ;;100) = ((𝐼._𝐽𝐾) · ;;100)
77	60, 64	mulcli 11343	. . 3 ⊢ ((𝐴.𝐵) · (𝐶.𝐷)) ∈ ℂ
78	47	nn0rei 12644	. . . . . 6 ⊢ 𝐽 ∈ ℝ
79		dp2cl 33516	. . . . . 6 ⊢ ((𝐽 ∈ ℝ ∧ 𝐾 ∈ ℝ) → _𝐽𝐾 ∈ ℝ)
80	78, 74, 79	mp2an 705	. . . . 5 ⊢ _𝐽𝐾 ∈ ℝ
81		dpcl 33527	. . . . 5 ⊢ ((𝐼 ∈ ℕ₀ ∧ _𝐽𝐾 ∈ ℝ) → (𝐼._𝐽𝐾) ∈ ℝ)
82	73, 80, 81	mp2an 705	. . . 4 ⊢ (𝐼._𝐽𝐾) ∈ ℝ
83	82	recni 11350	. . 3 ⊢ (𝐼._𝐽𝐾) ∈ ℂ
84		10nn 12861	. . . . . 6 ⊢ ;10 ∈ ℕ
85	84	decnncl2 12870	. . . . 5 ⊢ ;;100 ∈ ℕ
86	85	nncni 12372	. . . 4 ⊢ ;;100 ∈ ℂ
87	85	nnne0i 12405	. . . 4 ⊢ ;;100 ≠ 0
88	86, 87	pm3.2i 476	. . 3 ⊢ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)
89		mulcan2 11981	. . 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 1491	. 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

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 (class class class)co 7443 ℂcc 11225 ℝcr 11226 0cc0 11227 1c1 11228 + caddc 11230 · cmul 11232 ℕ₀cn0 12633 cdc 12841 cdp2 33507 .cdp 33524

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-er 8745 df-en 9002 df-dom 9003 df-sdom 9004 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-dec 12842 df-dp2 33508 df-dp 33525

This theorem is used by: hgt750lem2 35351

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