dpadd3 - Mathbox for Thierry Arnoux

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Theorem dpadd3 32879

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 12535	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
4		dpmul.c	. . . . . . . 8 ⊢ 𝐶 ∈ ℕ₀
5	4	nn0rei 12535	. . . . . . 7 ⊢ 𝐶 ∈ ℝ
6		dp2cl 32847	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ)
7	3, 5, 6	mp2an 692	. . . . . 6 ⊢ _𝐵𝐶 ∈ ℝ
8		dpcl 32858	. . . . . 6 ⊢ ((𝐴 ∈ ℕ₀ ∧ _𝐵𝐶 ∈ ℝ) → (𝐴._𝐵𝐶) ∈ ℝ)
9	1, 7, 8	mp2an 692	. . . . 5 ⊢ (𝐴._𝐵𝐶) ∈ ℝ
10	9	recni 11273	. . . 4 ⊢ (𝐴._𝐵𝐶) ∈ ℂ
11		dpmul.d	. . . . . 6 ⊢ 𝐷 ∈ ℕ₀
12		dpmul.e	. . . . . . . 8 ⊢ 𝐸 ∈ ℕ₀
13	12	nn0rei 12535	. . . . . . 7 ⊢ 𝐸 ∈ ℝ
14		dpadd3.f	. . . . . . . 8 ⊢ 𝐹 ∈ ℕ₀
15	14	nn0rei 12535	. . . . . . 7 ⊢ 𝐹 ∈ ℝ
16		dp2cl 32847	. . . . . . 7 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ)
17	13, 15, 16	mp2an 692	. . . . . 6 ⊢ _𝐸𝐹 ∈ ℝ
18		dpcl 32858	. . . . . 6 ⊢ ((𝐷 ∈ ℕ₀ ∧ _𝐸𝐹 ∈ ℝ) → (𝐷._𝐸𝐹) ∈ ℝ)
19	11, 17, 18	mp2an 692	. . . . 5 ⊢ (𝐷._𝐸𝐹) ∈ ℝ
20	19	recni 11273	. . . 4 ⊢ (𝐷._𝐸𝐹) ∈ ℂ
21	10, 20	addcli 11265	. . 3 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ
22		dpmul.g	. . . . 5 ⊢ 𝐺 ∈ ℕ₀
23		dpadd3.h	. . . . . . 7 ⊢ 𝐻 ∈ ℕ₀
24	23	nn0rei 12535	. . . . . 6 ⊢ 𝐻 ∈ ℝ
25		dpadd3.i	. . . . . . 7 ⊢ 𝐼 ∈ ℕ₀
26	25	nn0rei 12535	. . . . . 6 ⊢ 𝐼 ∈ ℝ
27		dp2cl 32847	. . . . . 6 ⊢ ((𝐻 ∈ ℝ ∧ 𝐼 ∈ ℝ) → _𝐻𝐼 ∈ ℝ)
28	24, 26, 27	mp2an 692	. . . . 5 ⊢ _𝐻𝐼 ∈ ℝ
29		dpcl 32858	. . . . 5 ⊢ ((𝐺 ∈ ℕ₀ ∧ _𝐻𝐼 ∈ ℝ) → (𝐺._𝐻𝐼) ∈ ℝ)
30	22, 28, 29	mp2an 692	. . . 4 ⊢ (𝐺._𝐻𝐼) ∈ ℝ
31	30	recni 11273	. . 3 ⊢ (𝐺._𝐻𝐼) ∈ ℂ
32		10nn 12747	. . . . . 6 ⊢ ;10 ∈ ℕ
33	32	decnncl2 12755	. . . . 5 ⊢ ;;100 ∈ ℕ
34	33	nncni 12274	. . . 4 ⊢ ;;100 ∈ ℂ
35	33	nnne0i 12304	. . . 4 ⊢ ;;100 ≠ 0
36	34, 35	pm3.2i 470	. . 3 ⊢ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)
37	21, 31, 36	3pm3.2i 1338	. 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ ∧ (𝐺._𝐻𝐼) ∈ ℂ ∧ (;;100 ∈ ℂ ∧ ;;100 ≠ 0))
38	10, 20, 34	adddiri 11272	. . 3 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100))
39		dpadd3.1	. . . 4 ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼
40	1, 2, 5	dpmul100 32864	. . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶
41	11, 12, 15	dpmul100 32864	. . . . 5 ⊢ ((𝐷._𝐸𝐹) · ;;100) = ;;𝐷𝐸𝐹
42	40, 41	oveq12i 7443	. . . 4 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹)
43	22, 23, 26	dpmul100 32864	. . . 4 ⊢ ((𝐺._𝐻𝐼) · ;;100) = ;;𝐺𝐻𝐼
44	39, 42, 43	3eqtr4i 2773	. . 3 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = ((𝐺._𝐻𝐼) · ;;100)
45	38, 44	eqtri 2763	. 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = ((𝐺._𝐻𝐼) · ;;100)
46		mulcan2 11899	. . 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 1537 ∈ wcel 2106 ≠ wne 2938 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ℕ₀cn0 12524 cdc 12731 cdp2 32838 .cdp 32855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-dec 12732 df-dp2 32839 df-dp 32856

This theorem is referenced by: 1mhdrd 32883 hgt750lem2 34646

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