dpadd2 - Mathbox for Thierry Arnoux

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Theorem dpadd2 33523

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 12621	. . . . 5 ⊢ 𝐴 ∈ ℝ
4		dpadd2.b	. . . . . 6 ⊢ 𝐵 ∈ ℝ⁺
5		rpre 13133	. . . . . 6 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 𝐵 ∈ ℝ
7		dp2cl 33493	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → _𝐴𝐵 ∈ ℝ)
8	3, 6, 7	mp2an 705	. . . 4 ⊢ _𝐴𝐵 ∈ ℝ
9	1, 8	dpval2 33506	. . 3 ⊢ (𝐺._𝐴𝐵) = (𝐺 + (_𝐴𝐵 / ;10))
10		dpadd2.h	. . . 4 ⊢ 𝐻 ∈ ℕ₀
11		dpadd2.c	. . . . . 6 ⊢ 𝐶 ∈ ℕ₀
12	11	nn0rei 12621	. . . . 5 ⊢ 𝐶 ∈ ℝ
13		dpadd2.d	. . . . . 6 ⊢ 𝐷 ∈ ℝ⁺
14		rpre 13133	. . . . . 6 ⊢ (𝐷 ∈ ℝ⁺ → 𝐷 ∈ ℝ)
15	13, 14	ax-mp 5	. . . . 5 ⊢ 𝐷 ∈ ℝ
16		dp2cl 33493	. . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ)
17	12, 15, 16	mp2an 705	. . . 4 ⊢ _𝐶𝐷 ∈ ℝ
18	10, 17	dpval2 33506	. . 3 ⊢ (𝐻._𝐶𝐷) = (𝐻 + (_𝐶𝐷 / ;10))
19	9, 18	oveq12i 7433	. 2 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = ((𝐺 + (_𝐴𝐵 / ;10)) + (𝐻 + (_𝐶𝐷 / ;10)))
20	1	nn0cni 12622	. . 3 ⊢ 𝐺 ∈ ℂ
21	8	recni 11327	. . . 4 ⊢ _𝐴𝐵 ∈ ℂ
22		10nn 12838	. . . . 5 ⊢ ;10 ∈ ℕ
23	22	nncni 12349	. . . 4 ⊢ ;10 ∈ ℂ
24	22	nnne0i 12382	. . . 4 ⊢ ;10 ≠ 0
25	21, 23, 24	divcli 12063	. . 3 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ
26	10	nn0cni 12622	. . 3 ⊢ 𝐻 ∈ ℂ
27	17	recni 11327	. . . 4 ⊢ _𝐶𝐷 ∈ ℂ
28	27, 23, 24	divcli 12063	. . 3 ⊢ (_𝐶𝐷 / ;10) ∈ ℂ
29	20, 25, 26, 28	add4i 11539	. 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 12078	. . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))
32		dpadd2.1	. . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)
33		dpval 33503	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)
34	2, 6, 33	mp2an 705	. . . . . . . 8 ⊢ (𝐴.𝐵) = _𝐴𝐵
35		dpval 33503	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷)
36	11, 15, 35	mp2an 705	. . . . . . . 8 ⊢ (𝐶.𝐷) = _𝐶𝐷
37	34, 36	oveq12i 7433	. . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (_𝐴𝐵 + _𝐶𝐷)
38		dpadd2.e	. . . . . . . 8 ⊢ 𝐸 ∈ ℕ₀
39		dpadd2.f	. . . . . . . . 9 ⊢ 𝐹 ∈ ℝ⁺
40		rpre 13133	. . . . . . . . 9 ⊢ (𝐹 ∈ ℝ⁺ → 𝐹 ∈ ℝ)
41	39, 40	ax-mp 5	. . . . . . . 8 ⊢ 𝐹 ∈ ℝ
42		dpval 33503	. . . . . . . 8 ⊢ ((𝐸 ∈ ℕ₀ ∧ 𝐹 ∈ ℝ) → (𝐸.𝐹) = _𝐸𝐹)
43	38, 41, 42	mp2an 705	. . . . . . 7 ⊢ (𝐸.𝐹) = _𝐸𝐹
44	32, 37, 43	3eqtr3i 2791	. . . . . 6 ⊢ (_𝐴𝐵 + _𝐶𝐷) = _𝐸𝐹
45	44	oveq1i 7431	. . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = (_𝐸𝐹 / ;10)
46	31, 45	eqtr3i 2785	. . . 4 ⊢ ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10)) = (_𝐸𝐹 / ;10)
47	30, 46	oveq12i 7433	. . 3 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼 + (_𝐸𝐹 / ;10))
48	1, 10	nn0addcli 12647	. . . . 5 ⊢ (𝐺 + 𝐻) ∈ ℕ₀
49	30, 48	eqeltrri 2857	. . . 4 ⊢ 𝐼 ∈ ℕ₀
50	38	nn0rei 12621	. . . . 5 ⊢ 𝐸 ∈ ℝ
51		dp2cl 33493	. . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ)
52	50, 41, 51	mp2an 705	. . . 4 ⊢ _𝐸𝐹 ∈ ℝ
53	49, 52	dpval2 33506	. . 3 ⊢ (𝐼._𝐸𝐹) = (𝐼 + (_𝐸𝐹 / ;10))
54	47, 53	eqtr4i 2786	. 2 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼._𝐸𝐹)
55	19, 29, 54	3eqtri 2787	1 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1569 ∈ wcel 2144 (class class class)co 7421 ℝcr 11203 0cc0 11204 1c1 11205 + caddc 11207 / cdiv 11977 ℕ₀cn0 12610 cdc 12818 ℝ⁺crp 13124 cdp2 33484 .cdp 33501

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-2nd 8003 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-er 8721 df-en 8978 df-dom 8979 df-sdom 8980 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-dec 12819 df-rp 13125 df-dp2 33485 df-dp 33502

This theorem is used by: hgt750lemd 35324

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