dpadd3 - Mathbox for Thierry Arnoux

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Theorem dpadd3 30761

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 11987	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
4		dpmul.c	. . . . . . . 8 ⊢ 𝐶 ∈ ℕ₀
5	4	nn0rei 11987	. . . . . . 7 ⊢ 𝐶 ∈ ℝ
6		dp2cl 30729	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ)
7	3, 5, 6	mp2an 692	. . . . . 6 ⊢ _𝐵𝐶 ∈ ℝ
8		dpcl 30740	. . . . . 6 ⊢ ((𝐴 ∈ ℕ₀ ∧ _𝐵𝐶 ∈ ℝ) → (𝐴._𝐵𝐶) ∈ ℝ)
9	1, 7, 8	mp2an 692	. . . . 5 ⊢ (𝐴._𝐵𝐶) ∈ ℝ
10	9	recni 10733	. . . 4 ⊢ (𝐴._𝐵𝐶) ∈ ℂ
11		dpmul.d	. . . . . 6 ⊢ 𝐷 ∈ ℕ₀
12		dpmul.e	. . . . . . . 8 ⊢ 𝐸 ∈ ℕ₀
13	12	nn0rei 11987	. . . . . . 7 ⊢ 𝐸 ∈ ℝ
14		dpadd3.f	. . . . . . . 8 ⊢ 𝐹 ∈ ℕ₀
15	14	nn0rei 11987	. . . . . . 7 ⊢ 𝐹 ∈ ℝ
16		dp2cl 30729	. . . . . . 7 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ)
17	13, 15, 16	mp2an 692	. . . . . 6 ⊢ _𝐸𝐹 ∈ ℝ
18		dpcl 30740	. . . . . 6 ⊢ ((𝐷 ∈ ℕ₀ ∧ _𝐸𝐹 ∈ ℝ) → (𝐷._𝐸𝐹) ∈ ℝ)
19	11, 17, 18	mp2an 692	. . . . 5 ⊢ (𝐷._𝐸𝐹) ∈ ℝ
20	19	recni 10733	. . . 4 ⊢ (𝐷._𝐸𝐹) ∈ ℂ
21	10, 20	addcli 10725	. . 3 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ
22		dpmul.g	. . . . 5 ⊢ 𝐺 ∈ ℕ₀
23		dpadd3.h	. . . . . . 7 ⊢ 𝐻 ∈ ℕ₀
24	23	nn0rei 11987	. . . . . 6 ⊢ 𝐻 ∈ ℝ
25		dpadd3.i	. . . . . . 7 ⊢ 𝐼 ∈ ℕ₀
26	25	nn0rei 11987	. . . . . 6 ⊢ 𝐼 ∈ ℝ
27		dp2cl 30729	. . . . . 6 ⊢ ((𝐻 ∈ ℝ ∧ 𝐼 ∈ ℝ) → _𝐻𝐼 ∈ ℝ)
28	24, 26, 27	mp2an 692	. . . . 5 ⊢ _𝐻𝐼 ∈ ℝ
29		dpcl 30740	. . . . 5 ⊢ ((𝐺 ∈ ℕ₀ ∧ _𝐻𝐼 ∈ ℝ) → (𝐺._𝐻𝐼) ∈ ℝ)
30	22, 28, 29	mp2an 692	. . . 4 ⊢ (𝐺._𝐻𝐼) ∈ ℝ
31	30	recni 10733	. . 3 ⊢ (𝐺._𝐻𝐼) ∈ ℂ
32		10nn 12195	. . . . . 6 ⊢ ;10 ∈ ℕ
33	32	decnncl2 12203	. . . . 5 ⊢ ;;100 ∈ ℕ
34	33	nncni 11726	. . . 4 ⊢ ;;100 ∈ ℂ
35	33	nnne0i 11756	. . . 4 ⊢ ;;100 ≠ 0
36	34, 35	pm3.2i 474	. . 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 10732	. . 3 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100))
39		dpadd3.1	. . . 4 ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼
40	1, 2, 5	dpmul100 30746	. . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶
41	11, 12, 15	dpmul100 30746	. . . . 5 ⊢ ((𝐷._𝐸𝐹) · ;;100) = ;;𝐷𝐸𝐹
42	40, 41	oveq12i 7182	. . . 4 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹)
43	22, 23, 26	dpmul100 30746	. . . 4 ⊢ ((𝐺._𝐻𝐼) · ;;100) = ;;𝐺𝐻𝐼
44	39, 42, 43	3eqtr4i 2771	. . 3 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = ((𝐺._𝐻𝐼) · ;;100)
45	38, 44	eqtri 2761	. 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = ((𝐺._𝐻𝐼) · ;;100)
46		mulcan2 11356	. . 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 480	. 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 (class class class)co 7170 ℂcc 10613 ℝcr 10614 0cc0 10615 1c1 10616 + caddc 10618 · cmul 10620 ℕ₀cn0 11976 cdc 12179 cdp2 30720 .cdp 30737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-dec 12180 df-dp2 30721 df-dp 30738

This theorem is referenced by: 1mhdrd 30765 hgt750lem2 32202

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