dpadd3 - Mathbox for Thierry Arnoux

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Theorem dpadd3 33574

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 12657	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
4		dpmul.c	. . . . . . . 8 ⊢ 𝐶 ∈ ℕ₀
5	4	nn0rei 12657	. . . . . . 7 ⊢ 𝐶 ∈ ℝ
6		dp2cl 33542	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ)
7	3, 5, 6	mp2an 705	. . . . . 6 ⊢ _𝐵𝐶 ∈ ℝ
8		dpcl 33553	. . . . . 6 ⊢ ((𝐴 ∈ ℕ₀ ∧ _𝐵𝐶 ∈ ℝ) → (𝐴._𝐵𝐶) ∈ ℝ)
9	1, 7, 8	mp2an 705	. . . . 5 ⊢ (𝐴._𝐵𝐶) ∈ ℝ
10	9	recni 11362	. . . 4 ⊢ (𝐴._𝐵𝐶) ∈ ℂ
11		dpmul.d	. . . . . 6 ⊢ 𝐷 ∈ ℕ₀
12		dpmul.e	. . . . . . . 8 ⊢ 𝐸 ∈ ℕ₀
13	12	nn0rei 12657	. . . . . . 7 ⊢ 𝐸 ∈ ℝ
14		dpadd3.f	. . . . . . . 8 ⊢ 𝐹 ∈ ℕ₀
15	14	nn0rei 12657	. . . . . . 7 ⊢ 𝐹 ∈ ℝ
16		dp2cl 33542	. . . . . . 7 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ)
17	13, 15, 16	mp2an 705	. . . . . 6 ⊢ _𝐸𝐹 ∈ ℝ
18		dpcl 33553	. . . . . 6 ⊢ ((𝐷 ∈ ℕ₀ ∧ _𝐸𝐹 ∈ ℝ) → (𝐷._𝐸𝐹) ∈ ℝ)
19	11, 17, 18	mp2an 705	. . . . 5 ⊢ (𝐷._𝐸𝐹) ∈ ℝ
20	19	recni 11362	. . . 4 ⊢ (𝐷._𝐸𝐹) ∈ ℂ
21	10, 20	addcli 11354	. . 3 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ
22		dpmul.g	. . . . 5 ⊢ 𝐺 ∈ ℕ₀
23		dpadd3.h	. . . . . . 7 ⊢ 𝐻 ∈ ℕ₀
24	23	nn0rei 12657	. . . . . 6 ⊢ 𝐻 ∈ ℝ
25		dpadd3.i	. . . . . . 7 ⊢ 𝐼 ∈ ℕ₀
26	25	nn0rei 12657	. . . . . 6 ⊢ 𝐼 ∈ ℝ
27		dp2cl 33542	. . . . . 6 ⊢ ((𝐻 ∈ ℝ ∧ 𝐼 ∈ ℝ) → _𝐻𝐼 ∈ ℝ)
28	24, 26, 27	mp2an 705	. . . . 5 ⊢ _𝐻𝐼 ∈ ℝ
29		dpcl 33553	. . . . 5 ⊢ ((𝐺 ∈ ℕ₀ ∧ _𝐻𝐼 ∈ ℝ) → (𝐺._𝐻𝐼) ∈ ℝ)
30	22, 28, 29	mp2an 705	. . . 4 ⊢ (𝐺._𝐻𝐼) ∈ ℝ
31	30	recni 11362	. . 3 ⊢ (𝐺._𝐻𝐼) ∈ ℂ
32		10nn 12874	. . . . . 6 ⊢ ;10 ∈ ℕ
33	32	decnncl2 12883	. . . . 5 ⊢ ;;100 ∈ ℕ
34	33	nncni 12385	. . . 4 ⊢ ;;100 ∈ ℂ
35	33	nnne0i 12418	. . . 4 ⊢ ;;100 ≠ 0
36	34, 35	pm3.2i 476	. . 3 ⊢ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)
37	21, 31, 36	3pm3.2i 1358	. 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ ∧ (𝐺._𝐻𝐼) ∈ ℂ ∧ (;;100 ∈ ℂ ∧ ;;100 ≠ 0))
38	10, 20, 34	adddiri 11361	. . 3 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100))
39		dpadd3.1	. . . 4 ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼
40	1, 2, 5	dpmul100 33559	. . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶
41	11, 12, 15	dpmul100 33559	. . . . 5 ⊢ ((𝐷._𝐸𝐹) · ;;100) = ;;𝐷𝐸𝐹
42	40, 41	oveq12i 7463	. . . 4 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹)
43	22, 23, 26	dpmul100 33559	. . . 4 ⊢ ((𝐺._𝐻𝐼) · ;;100) = ;;𝐺𝐻𝐼
44	39, 42, 43	3eqtr4i 2796	. . 3 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = ((𝐺._𝐻𝐼) · ;;100)
45	38, 44	eqtri 2786	. 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = ((𝐺._𝐻𝐼) · ;;100)
46		mulcan2 11994	. . 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 482	. 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 705	1 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 + caddc 11242 · cmul 11244 ℕ₀cn0 12646 cdc 12854 cdp2 33533 .cdp 33550

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-dec 12855 df-dp2 33534 df-dp 33551

This theorem is used by: 1mhdrd 33578 hgt750lem2 35392

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