dpadd3 - Mathbox for Thierry Arnoux

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Theorem dpadd3 32832

Description: Addition with two decimals. (Contributed by Thierry Arnoux, 27-Dec-2021.)

**Hypotheses**
Ref	Expression
dpmul.a	⊢ 𝐴 ∈ ℕ₀
dpmul.b	⊢ 𝐵 ∈ ℕ₀
dpmul.c	⊢ 𝐶 ∈ ℕ₀
dpmul.d	⊢ 𝐷 ∈ ℕ₀
dpmul.e	⊢ 𝐸 ∈ ℕ₀
dpmul.g	⊢ 𝐺 ∈ ℕ₀
dpadd3.f	⊢ 𝐹 ∈ ℕ₀
dpadd3.h	⊢ 𝐻 ∈ ℕ₀
dpadd3.i	⊢ 𝐼 ∈ ℕ₀
dpadd3.1	⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼

**Assertion**
Ref	Expression
dpadd3	⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼)

**Proof of Theorem dpadd3**
Step	Hyp	Ref	Expression
1		dpmul.a	. . . . . 6 ⊢ 𝐴 ∈ ℕ₀
2		dpmul.b	. . . . . . . 8 ⊢ 𝐵 ∈ ℕ₀
3	2	nn0rei 12453	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
4		dpmul.c	. . . . . . . 8 ⊢ 𝐶 ∈ ℕ₀
5	4	nn0rei 12453	. . . . . . 7 ⊢ 𝐶 ∈ ℝ
6		dp2cl 32800	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ)
7	3, 5, 6	mp2an 692	. . . . . 6 ⊢ _𝐵𝐶 ∈ ℝ
8		dpcl 32811	. . . . . 6 ⊢ ((𝐴 ∈ ℕ₀ ∧ _𝐵𝐶 ∈ ℝ) → (𝐴._𝐵𝐶) ∈ ℝ)
9	1, 7, 8	mp2an 692	. . . . 5 ⊢ (𝐴._𝐵𝐶) ∈ ℝ
10	9	recni 11188	. . . 4 ⊢ (𝐴._𝐵𝐶) ∈ ℂ
11		dpmul.d	. . . . . 6 ⊢ 𝐷 ∈ ℕ₀
12		dpmul.e	. . . . . . . 8 ⊢ 𝐸 ∈ ℕ₀
13	12	nn0rei 12453	. . . . . . 7 ⊢ 𝐸 ∈ ℝ
14		dpadd3.f	. . . . . . . 8 ⊢ 𝐹 ∈ ℕ₀
15	14	nn0rei 12453	. . . . . . 7 ⊢ 𝐹 ∈ ℝ
16		dp2cl 32800	. . . . . . 7 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ)
17	13, 15, 16	mp2an 692	. . . . . 6 ⊢ _𝐸𝐹 ∈ ℝ
18		dpcl 32811	. . . . . 6 ⊢ ((𝐷 ∈ ℕ₀ ∧ _𝐸𝐹 ∈ ℝ) → (𝐷._𝐸𝐹) ∈ ℝ)
19	11, 17, 18	mp2an 692	. . . . 5 ⊢ (𝐷._𝐸𝐹) ∈ ℝ
20	19	recni 11188	. . . 4 ⊢ (𝐷._𝐸𝐹) ∈ ℂ
21	10, 20	addcli 11180	. . 3 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ
22		dpmul.g	. . . . 5 ⊢ 𝐺 ∈ ℕ₀
23		dpadd3.h	. . . . . . 7 ⊢ 𝐻 ∈ ℕ₀
24	23	nn0rei 12453	. . . . . 6 ⊢ 𝐻 ∈ ℝ
25		dpadd3.i	. . . . . . 7 ⊢ 𝐼 ∈ ℕ₀
26	25	nn0rei 12453	. . . . . 6 ⊢ 𝐼 ∈ ℝ
27		dp2cl 32800	. . . . . 6 ⊢ ((𝐻 ∈ ℝ ∧ 𝐼 ∈ ℝ) → _𝐻𝐼 ∈ ℝ)
28	24, 26, 27	mp2an 692	. . . . 5 ⊢ _𝐻𝐼 ∈ ℝ
29		dpcl 32811	. . . . 5 ⊢ ((𝐺 ∈ ℕ₀ ∧ _𝐻𝐼 ∈ ℝ) → (𝐺._𝐻𝐼) ∈ ℝ)
30	22, 28, 29	mp2an 692	. . . 4 ⊢ (𝐺._𝐻𝐼) ∈ ℝ
31	30	recni 11188	. . 3 ⊢ (𝐺._𝐻𝐼) ∈ ℂ
32		10nn 12665	. . . . . 6 ⊢ ;10 ∈ ℕ
33	32	decnncl2 12673	. . . . 5 ⊢ ;;100 ∈ ℕ
34	33	nncni 12196	. . . 4 ⊢ ;;100 ∈ ℂ
35	33	nnne0i 12226	. . . 4 ⊢ ;;100 ≠ 0
36	34, 35	pm3.2i 470	. . 3 ⊢ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)
37	21, 31, 36	3pm3.2i 1340	. 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ ∧ (𝐺._𝐻𝐼) ∈ ℂ ∧ (;;100 ∈ ℂ ∧ ;;100 ≠ 0))
38	10, 20, 34	adddiri 11187	. . 3 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100))
39		dpadd3.1	. . . 4 ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼
40	1, 2, 5	dpmul100 32817	. . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶
41	11, 12, 15	dpmul100 32817	. . . . 5 ⊢ ((𝐷._𝐸𝐹) · ;;100) = ;;𝐷𝐸𝐹
42	40, 41	oveq12i 7399	. . . 4 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹)
43	22, 23, 26	dpmul100 32817	. . . 4 ⊢ ((𝐺._𝐻𝐼) · ;;100) = ;;𝐺𝐻𝐼
44	39, 42, 43	3eqtr4i 2762	. . 3 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = ((𝐺._𝐻𝐼) · ;;100)
45	38, 44	eqtri 2752	. 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = ((𝐺._𝐻𝐼) · ;;100)
46		mulcan2 11816	. . 3 ⊢ ((((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ ∧ (𝐺._𝐻𝐼) ∈ ℂ ∧ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)) → ((((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = ((𝐺._𝐻𝐼) · ;;100) ↔ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼)))
47	46	biimpa 476	. 2 ⊢ (((((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ ∧ (𝐺._𝐻𝐼) ∈ ℂ ∧ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)) ∧ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = ((𝐺._𝐻𝐼) · ;;100)) → ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼))
48	37, 45, 47	mp2an 692	1 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 ℕ₀cn0 12442 cdc 12649 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-dp2 32792 df-dp 32809

This theorem is referenced by: 1mhdrd 32836 hgt750lem2 34643

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