dpadd2 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dpadd2		Structured version Visualization version GIF version

Theorem dpadd2 32076

Description: Addition with one decimal, no carry. (Contributed by Thierry Arnoux, 29-Dec-2021.)

**Hypotheses**
Ref	Expression
dpadd2.a	⊢ 𝐴 ∈ ℕ₀
dpadd2.b	⊢ 𝐵 ∈ ℝ⁺
dpadd2.c	⊢ 𝐶 ∈ ℕ₀
dpadd2.d	⊢ 𝐷 ∈ ℝ⁺
dpadd2.e	⊢ 𝐸 ∈ ℕ₀
dpadd2.f	⊢ 𝐹 ∈ ℝ⁺
dpadd2.g	⊢ 𝐺 ∈ ℕ₀
dpadd2.h	⊢ 𝐻 ∈ ℕ₀
dpadd2.i	⊢ (𝐺 + 𝐻) = 𝐼
dpadd2.1	⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)

**Assertion**
Ref	Expression
dpadd2	⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹)

**Proof of Theorem dpadd2**
Step	Hyp	Ref	Expression
1		dpadd2.g	. . . 4 ⊢ 𝐺 ∈ ℕ₀
2		dpadd2.a	. . . . . 6 ⊢ 𝐴 ∈ ℕ₀
3	2	nn0rei 12483	. . . . 5 ⊢ 𝐴 ∈ ℝ
4		dpadd2.b	. . . . . 6 ⊢ 𝐵 ∈ ℝ⁺
5		rpre 12982	. . . . . 6 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 𝐵 ∈ ℝ
7		dp2cl 32046	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → _𝐴𝐵 ∈ ℝ)
8	3, 6, 7	mp2an 691	. . . 4 ⊢ _𝐴𝐵 ∈ ℝ
9	1, 8	dpval2 32059	. . 3 ⊢ (𝐺._𝐴𝐵) = (𝐺 + (_𝐴𝐵 / ;10))
10		dpadd2.h	. . . 4 ⊢ 𝐻 ∈ ℕ₀
11		dpadd2.c	. . . . . 6 ⊢ 𝐶 ∈ ℕ₀
12	11	nn0rei 12483	. . . . 5 ⊢ 𝐶 ∈ ℝ
13		dpadd2.d	. . . . . 6 ⊢ 𝐷 ∈ ℝ⁺
14		rpre 12982	. . . . . 6 ⊢ (𝐷 ∈ ℝ⁺ → 𝐷 ∈ ℝ)
15	13, 14	ax-mp 5	. . . . 5 ⊢ 𝐷 ∈ ℝ
16		dp2cl 32046	. . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ)
17	12, 15, 16	mp2an 691	. . . 4 ⊢ _𝐶𝐷 ∈ ℝ
18	10, 17	dpval2 32059	. . 3 ⊢ (𝐻._𝐶𝐷) = (𝐻 + (_𝐶𝐷 / ;10))
19	9, 18	oveq12i 7421	. 2 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = ((𝐺 + (_𝐴𝐵 / ;10)) + (𝐻 + (_𝐶𝐷 / ;10)))
20	1	nn0cni 12484	. . 3 ⊢ 𝐺 ∈ ℂ
21	8	recni 11228	. . . 4 ⊢ _𝐴𝐵 ∈ ℂ
22		10nn 12693	. . . . 5 ⊢ ;10 ∈ ℕ
23	22	nncni 12222	. . . 4 ⊢ ;10 ∈ ℂ
24	22	nnne0i 12252	. . . 4 ⊢ ;10 ≠ 0
25	21, 23, 24	divcli 11956	. . 3 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ
26	10	nn0cni 12484	. . 3 ⊢ 𝐻 ∈ ℂ
27	17	recni 11228	. . . 4 ⊢ _𝐶𝐷 ∈ ℂ
28	27, 23, 24	divcli 11956	. . 3 ⊢ (_𝐶𝐷 / ;10) ∈ ℂ
29	20, 25, 26, 28	add4i 11438	. 2 ⊢ ((𝐺 + (_𝐴𝐵 / ;10)) + (𝐻 + (_𝐶𝐷 / ;10))) = ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10)))
30		dpadd2.i	. . . 4 ⊢ (𝐺 + 𝐻) = 𝐼
31	21, 27, 23, 24	divdiri 11971	. . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))
32		dpadd2.1	. . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)
33		dpval 32056	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)
34	2, 6, 33	mp2an 691	. . . . . . . 8 ⊢ (𝐴.𝐵) = _𝐴𝐵
35		dpval 32056	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷)
36	11, 15, 35	mp2an 691	. . . . . . . 8 ⊢ (𝐶.𝐷) = _𝐶𝐷
37	34, 36	oveq12i 7421	. . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (_𝐴𝐵 + _𝐶𝐷)
38		dpadd2.e	. . . . . . . 8 ⊢ 𝐸 ∈ ℕ₀
39		dpadd2.f	. . . . . . . . 9 ⊢ 𝐹 ∈ ℝ⁺
40		rpre 12982	. . . . . . . . 9 ⊢ (𝐹 ∈ ℝ⁺ → 𝐹 ∈ ℝ)
41	39, 40	ax-mp 5	. . . . . . . 8 ⊢ 𝐹 ∈ ℝ
42		dpval 32056	. . . . . . . 8 ⊢ ((𝐸 ∈ ℕ₀ ∧ 𝐹 ∈ ℝ) → (𝐸.𝐹) = _𝐸𝐹)
43	38, 41, 42	mp2an 691	. . . . . . 7 ⊢ (𝐸.𝐹) = _𝐸𝐹
44	32, 37, 43	3eqtr3i 2769	. . . . . 6 ⊢ (_𝐴𝐵 + _𝐶𝐷) = _𝐸𝐹
45	44	oveq1i 7419	. . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = (_𝐸𝐹 / ;10)
46	31, 45	eqtr3i 2763	. . . 4 ⊢ ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10)) = (_𝐸𝐹 / ;10)
47	30, 46	oveq12i 7421	. . 3 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼 + (_𝐸𝐹 / ;10))
48	1, 10	nn0addcli 12509	. . . . 5 ⊢ (𝐺 + 𝐻) ∈ ℕ₀
49	30, 48	eqeltrri 2831	. . . 4 ⊢ 𝐼 ∈ ℕ₀
50	38	nn0rei 12483	. . . . 5 ⊢ 𝐸 ∈ ℝ
51		dp2cl 32046	. . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ)
52	50, 41, 51	mp2an 691	. . . 4 ⊢ _𝐸𝐹 ∈ ℝ
53	49, 52	dpval2 32059	. . 3 ⊢ (𝐼._𝐸𝐹) = (𝐼 + (_𝐸𝐹 / ;10))
54	47, 53	eqtr4i 2764	. 2 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼._𝐸𝐹)
55	19, 29, 54	3eqtri 2765	1 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹)

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7409 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 / cdiv 11871 ℕ₀cn0 12472 cdc 12677 ℝ⁺crp 12974 cdp2 32037 .cdp 32054

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-dec 12678 df-rp 12975 df-dp2 32038 df-dp 32055

This theorem is referenced by: hgt750lemd 33660

W3C validator