dpadd2 - Mathbox for Thierry Arnoux

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Theorem dpadd2 33266

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 12533	. . . . 5 ⊢ 𝐴 ∈ ℝ
4		dpadd2.b	. . . . . 6 ⊢ 𝐵 ∈ ℝ⁺
5		rpre 13043	. . . . . 6 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 𝐵 ∈ ℝ
7		dp2cl 33236	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → _𝐴𝐵 ∈ ℝ)
8	3, 6, 7	mp2an 705	. . . 4 ⊢ _𝐴𝐵 ∈ ℝ
9	1, 8	dpval2 33249	. . 3 ⊢ (𝐺._𝐴𝐵) = (𝐺 + (_𝐴𝐵 / ;10))
10		dpadd2.h	. . . 4 ⊢ 𝐻 ∈ ℕ₀
11		dpadd2.c	. . . . . 6 ⊢ 𝐶 ∈ ℕ₀
12	11	nn0rei 12533	. . . . 5 ⊢ 𝐶 ∈ ℝ
13		dpadd2.d	. . . . . 6 ⊢ 𝐷 ∈ ℝ⁺
14		rpre 13043	. . . . . 6 ⊢ (𝐷 ∈ ℝ⁺ → 𝐷 ∈ ℝ)
15	13, 14	ax-mp 5	. . . . 5 ⊢ 𝐷 ∈ ℝ
16		dp2cl 33236	. . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ)
17	12, 15, 16	mp2an 705	. . . 4 ⊢ _𝐶𝐷 ∈ ℝ
18	10, 17	dpval2 33249	. . 3 ⊢ (𝐻._𝐶𝐷) = (𝐻 + (_𝐶𝐷 / ;10))
19	9, 18	oveq12i 7435	. 2 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = ((𝐺 + (_𝐴𝐵 / ;10)) + (𝐻 + (_𝐶𝐷 / ;10)))
20	1	nn0cni 12534	. . 3 ⊢ 𝐺 ∈ ℂ
21	8	recni 11241	. . . 4 ⊢ _𝐴𝐵 ∈ ℂ
22		10nn 12749	. . . . 5 ⊢ ;10 ∈ ℕ
23	22	nncni 12261	. . . 4 ⊢ ;10 ∈ ℂ
24	22	nnne0i 12294	. . . 4 ⊢ ;10 ≠ 0
25	21, 23, 24	divcli 11975	. . 3 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ
26	10	nn0cni 12534	. . 3 ⊢ 𝐻 ∈ ℂ
27	17	recni 11241	. . . 4 ⊢ _𝐶𝐷 ∈ ℂ
28	27, 23, 24	divcli 11975	. . 3 ⊢ (_𝐶𝐷 / ;10) ∈ ℂ
29	20, 25, 26, 28	add4i 11453	. 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 11990	. . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))
32		dpadd2.1	. . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)
33		dpval 33246	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)
34	2, 6, 33	mp2an 705	. . . . . . . 8 ⊢ (𝐴.𝐵) = _𝐴𝐵
35		dpval 33246	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷)
36	11, 15, 35	mp2an 705	. . . . . . . 8 ⊢ (𝐶.𝐷) = _𝐶𝐷
37	34, 36	oveq12i 7435	. . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (_𝐴𝐵 + _𝐶𝐷)
38		dpadd2.e	. . . . . . . 8 ⊢ 𝐸 ∈ ℕ₀
39		dpadd2.f	. . . . . . . . 9 ⊢ 𝐹 ∈ ℝ⁺
40		rpre 13043	. . . . . . . . 9 ⊢ (𝐹 ∈ ℝ⁺ → 𝐹 ∈ ℝ)
41	39, 40	ax-mp 5	. . . . . . . 8 ⊢ 𝐹 ∈ ℝ
42		dpval 33246	. . . . . . . 8 ⊢ ((𝐸 ∈ ℕ₀ ∧ 𝐹 ∈ ℝ) → (𝐸.𝐹) = _𝐸𝐹)
43	38, 41, 42	mp2an 705	. . . . . . 7 ⊢ (𝐸.𝐹) = _𝐸𝐹
44	32, 37, 43	3eqtr3i 2797	. . . . . 6 ⊢ (_𝐴𝐵 + _𝐶𝐷) = _𝐸𝐹
45	44	oveq1i 7433	. . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = (_𝐸𝐹 / ;10)
46	31, 45	eqtr3i 2791	. . . 4 ⊢ ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10)) = (_𝐸𝐹 / ;10)
47	30, 46	oveq12i 7435	. . 3 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼 + (_𝐸𝐹 / ;10))
48	1, 10	nn0addcli 12559	. . . . 5 ⊢ (𝐺 + 𝐻) ∈ ℕ₀
49	30, 48	eqeltrri 2863	. . . 4 ⊢ 𝐼 ∈ ℕ₀
50	38	nn0rei 12533	. . . . 5 ⊢ 𝐸 ∈ ℝ
51		dp2cl 33236	. . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ)
52	50, 41, 51	mp2an 705	. . . 4 ⊢ _𝐸𝐹 ∈ ℝ
53	49, 52	dpval2 33249	. . 3 ⊢ (𝐼._𝐸𝐹) = (𝐼 + (_𝐸𝐹 / ;10))
54	47, 53	eqtr4i 2792	. 2 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼._𝐸𝐹)
55	19, 29, 54	3eqtri 2793	1 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2146 (class class class)co 7423 ℝcr 11117 0cc0 11118 1c1 11119 + caddc 11121 / cdiv 11889 ℕ₀cn0 12522 cdc 12729 ℝ⁺crp 13034 cdp2 33227 .cdp 33244

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

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 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-dec 12730 df-rp 13035 df-dp2 33228 df-dp 33245

This theorem is used by: hgt750lemd 35067

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