dpadd2 - Mathbox for Thierry Arnoux

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Theorem dpadd2 32830

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 12453	. . . . 5 ⊢ 𝐴 ∈ ℝ
4		dpadd2.b	. . . . . 6 ⊢ 𝐵 ∈ ℝ⁺
5		rpre 12960	. . . . . 6 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 𝐵 ∈ ℝ
7		dp2cl 32800	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → _𝐴𝐵 ∈ ℝ)
8	3, 6, 7	mp2an 692	. . . 4 ⊢ _𝐴𝐵 ∈ ℝ
9	1, 8	dpval2 32813	. . 3 ⊢ (𝐺._𝐴𝐵) = (𝐺 + (_𝐴𝐵 / ;10))
10		dpadd2.h	. . . 4 ⊢ 𝐻 ∈ ℕ₀
11		dpadd2.c	. . . . . 6 ⊢ 𝐶 ∈ ℕ₀
12	11	nn0rei 12453	. . . . 5 ⊢ 𝐶 ∈ ℝ
13		dpadd2.d	. . . . . 6 ⊢ 𝐷 ∈ ℝ⁺
14		rpre 12960	. . . . . 6 ⊢ (𝐷 ∈ ℝ⁺ → 𝐷 ∈ ℝ)
15	13, 14	ax-mp 5	. . . . 5 ⊢ 𝐷 ∈ ℝ
16		dp2cl 32800	. . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ)
17	12, 15, 16	mp2an 692	. . . 4 ⊢ _𝐶𝐷 ∈ ℝ
18	10, 17	dpval2 32813	. . 3 ⊢ (𝐻._𝐶𝐷) = (𝐻 + (_𝐶𝐷 / ;10))
19	9, 18	oveq12i 7399	. 2 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = ((𝐺 + (_𝐴𝐵 / ;10)) + (𝐻 + (_𝐶𝐷 / ;10)))
20	1	nn0cni 12454	. . 3 ⊢ 𝐺 ∈ ℂ
21	8	recni 11188	. . . 4 ⊢ _𝐴𝐵 ∈ ℂ
22		10nn 12665	. . . . 5 ⊢ ;10 ∈ ℕ
23	22	nncni 12196	. . . 4 ⊢ ;10 ∈ ℂ
24	22	nnne0i 12226	. . . 4 ⊢ ;10 ≠ 0
25	21, 23, 24	divcli 11924	. . 3 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ
26	10	nn0cni 12454	. . 3 ⊢ 𝐻 ∈ ℂ
27	17	recni 11188	. . . 4 ⊢ _𝐶𝐷 ∈ ℂ
28	27, 23, 24	divcli 11924	. . 3 ⊢ (_𝐶𝐷 / ;10) ∈ ℂ
29	20, 25, 26, 28	add4i 11399	. 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 11939	. . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))
32		dpadd2.1	. . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)
33		dpval 32810	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)
34	2, 6, 33	mp2an 692	. . . . . . . 8 ⊢ (𝐴.𝐵) = _𝐴𝐵
35		dpval 32810	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷)
36	11, 15, 35	mp2an 692	. . . . . . . 8 ⊢ (𝐶.𝐷) = _𝐶𝐷
37	34, 36	oveq12i 7399	. . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (_𝐴𝐵 + _𝐶𝐷)
38		dpadd2.e	. . . . . . . 8 ⊢ 𝐸 ∈ ℕ₀
39		dpadd2.f	. . . . . . . . 9 ⊢ 𝐹 ∈ ℝ⁺
40		rpre 12960	. . . . . . . . 9 ⊢ (𝐹 ∈ ℝ⁺ → 𝐹 ∈ ℝ)
41	39, 40	ax-mp 5	. . . . . . . 8 ⊢ 𝐹 ∈ ℝ
42		dpval 32810	. . . . . . . 8 ⊢ ((𝐸 ∈ ℕ₀ ∧ 𝐹 ∈ ℝ) → (𝐸.𝐹) = _𝐸𝐹)
43	38, 41, 42	mp2an 692	. . . . . . 7 ⊢ (𝐸.𝐹) = _𝐸𝐹
44	32, 37, 43	3eqtr3i 2760	. . . . . 6 ⊢ (_𝐴𝐵 + _𝐶𝐷) = _𝐸𝐹
45	44	oveq1i 7397	. . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = (_𝐸𝐹 / ;10)
46	31, 45	eqtr3i 2754	. . . 4 ⊢ ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10)) = (_𝐸𝐹 / ;10)
47	30, 46	oveq12i 7399	. . 3 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼 + (_𝐸𝐹 / ;10))
48	1, 10	nn0addcli 12479	. . . . 5 ⊢ (𝐺 + 𝐻) ∈ ℕ₀
49	30, 48	eqeltrri 2825	. . . 4 ⊢ 𝐼 ∈ ℕ₀
50	38	nn0rei 12453	. . . . 5 ⊢ 𝐸 ∈ ℝ
51		dp2cl 32800	. . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ)
52	50, 41, 51	mp2an 692	. . . 4 ⊢ _𝐸𝐹 ∈ ℝ
53	49, 52	dpval2 32813	. . 3 ⊢ (𝐼._𝐸𝐹) = (𝐼 + (_𝐸𝐹 / ;10))
54	47, 53	eqtr4i 2755	. 2 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼._𝐸𝐹)
55	19, 29, 54	3eqtri 2756	1 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹)

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 (class class class)co 7387 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 / cdiv 11835 ℕ₀cn0 12442 cdc 12649 ℝ⁺crp 12951 cdp2 32791 .cdp 32808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-dec 12650 df-rp 12952 df-dp2 32792 df-dp 32809

This theorem is referenced by: hgt750lemd 34639

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