dpadd2 - Mathbox for Thierry Arnoux

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Theorem dpadd2 29618

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 11303	. . . . 5 ⊢ 𝐴 ∈ ℝ
4		dpadd2.b	. . . . . 6 ⊢ 𝐵 ∈ ℝ⁺
5		rpre 11839	. . . . . 6 ⊢ (𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 𝐵 ∈ ℝ
7		dp2cl 29587	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → _𝐴𝐵 ∈ ℝ)
8	3, 6, 7	mp2an 708	. . . 4 ⊢ _𝐴𝐵 ∈ ℝ
9	1, 8	dpval2 29601	. . 3 ⊢ (𝐺._𝐴𝐵) = (𝐺 + (_𝐴𝐵 / ;10))
10		dpadd2.h	. . . 4 ⊢ 𝐻 ∈ ℕ₀
11		dpadd2.c	. . . . . 6 ⊢ 𝐶 ∈ ℕ₀
12	11	nn0rei 11303	. . . . 5 ⊢ 𝐶 ∈ ℝ
13		dpadd2.d	. . . . . 6 ⊢ 𝐷 ∈ ℝ⁺
14		rpre 11839	. . . . . 6 ⊢ (𝐷 ∈ ℝ⁺ → 𝐷 ∈ ℝ)
15	13, 14	ax-mp 5	. . . . 5 ⊢ 𝐷 ∈ ℝ
16		dp2cl 29587	. . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ)
17	12, 15, 16	mp2an 708	. . . 4 ⊢ _𝐶𝐷 ∈ ℝ
18	10, 17	dpval2 29601	. . 3 ⊢ (𝐻._𝐶𝐷) = (𝐻 + (_𝐶𝐷 / ;10))
19	9, 18	oveq12i 6662	. 2 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = ((𝐺 + (_𝐴𝐵 / ;10)) + (𝐻 + (_𝐶𝐷 / ;10)))
20	1	nn0cni 11304	. . 3 ⊢ 𝐺 ∈ ℂ
21	8	recni 10052	. . . 4 ⊢ _𝐴𝐵 ∈ ℂ
22		10nn 11514	. . . . 5 ⊢ ;10 ∈ ℕ
23	22	nncni 11030	. . . 4 ⊢ ;10 ∈ ℂ
24	22	nnne0i 11055	. . . 4 ⊢ ;10 ≠ 0
25	21, 23, 24	divcli 10767	. . 3 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ
26	10	nn0cni 11304	. . 3 ⊢ 𝐻 ∈ ℂ
27	17	recni 10052	. . . 4 ⊢ _𝐶𝐷 ∈ ℂ
28	27, 23, 24	divcli 10767	. . 3 ⊢ (_𝐶𝐷 / ;10) ∈ ℂ
29	20, 25, 26, 28	add4i 10260	. 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 10782	. . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))
32		dpadd2.1	. . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹)
33		dpval 29597	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵)
34	2, 6, 33	mp2an 708	. . . . . . . 8 ⊢ (𝐴.𝐵) = _𝐴𝐵
35		dpval 29597	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷)
36	11, 15, 35	mp2an 708	. . . . . . . 8 ⊢ (𝐶.𝐷) = _𝐶𝐷
37	34, 36	oveq12i 6662	. . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (_𝐴𝐵 + _𝐶𝐷)
38		dpadd2.e	. . . . . . . 8 ⊢ 𝐸 ∈ ℕ₀
39		dpadd2.f	. . . . . . . . 9 ⊢ 𝐹 ∈ ℝ⁺
40		rpre 11839	. . . . . . . . 9 ⊢ (𝐹 ∈ ℝ⁺ → 𝐹 ∈ ℝ)
41	39, 40	ax-mp 5	. . . . . . . 8 ⊢ 𝐹 ∈ ℝ
42		dpval 29597	. . . . . . . 8 ⊢ ((𝐸 ∈ ℕ₀ ∧ 𝐹 ∈ ℝ) → (𝐸.𝐹) = _𝐸𝐹)
43	38, 41, 42	mp2an 708	. . . . . . 7 ⊢ (𝐸.𝐹) = _𝐸𝐹
44	32, 37, 43	3eqtr3i 2652	. . . . . 6 ⊢ (_𝐴𝐵 + _𝐶𝐷) = _𝐸𝐹
45	44	oveq1i 6660	. . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = (_𝐸𝐹 / ;10)
46	31, 45	eqtr3i 2646	. . . 4 ⊢ ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10)) = (_𝐸𝐹 / ;10)
47	30, 46	oveq12i 6662	. . 3 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼 + (_𝐸𝐹 / ;10))
48	1, 10	nn0addcli 11330	. . . . 5 ⊢ (𝐺 + 𝐻) ∈ ℕ₀
49	30, 48	eqeltrri 2698	. . . 4 ⊢ 𝐼 ∈ ℕ₀
50	38	nn0rei 11303	. . . . 5 ⊢ 𝐸 ∈ ℝ
51		dp2cl 29587	. . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ)
52	50, 41, 51	mp2an 708	. . . 4 ⊢ _𝐸𝐹 ∈ ℝ
53	49, 52	dpval2 29601	. . 3 ⊢ (𝐼._𝐸𝐹) = (𝐼 + (_𝐸𝐹 / ;10))
54	47, 53	eqtr4i 2647	. 2 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼._𝐸𝐹)
55	19, 29, 54	3eqtri 2648	1 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹)

Colors of variables: wff setvar class

Syntax hints: = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 / cdiv 10684 ℕ₀cn0 11292 cdc 11493 ℝ⁺crp 11832 cdp2 29577 .cdp 29595

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-dec 11494 df-rp 11833 df-dp2 29578 df-dp 29596

This theorem is referenced by: hgt750lemd 30726

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