dpadd3 - Mathbox for Thierry Arnoux

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Theorem dpadd3 32865

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 12413	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
4		dpmul.c	. . . . . . . 8 ⊢ 𝐶 ∈ ℕ₀
5	4	nn0rei 12413	. . . . . . 7 ⊢ 𝐶 ∈ ℝ
6		dp2cl 32833	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ)
7	3, 5, 6	mp2an 692	. . . . . 6 ⊢ _𝐵𝐶 ∈ ℝ
8		dpcl 32844	. . . . . 6 ⊢ ((𝐴 ∈ ℕ₀ ∧ _𝐵𝐶 ∈ ℝ) → (𝐴._𝐵𝐶) ∈ ℝ)
9	1, 7, 8	mp2an 692	. . . . 5 ⊢ (𝐴._𝐵𝐶) ∈ ℝ
10	9	recni 11148	. . . 4 ⊢ (𝐴._𝐵𝐶) ∈ ℂ
11		dpmul.d	. . . . . 6 ⊢ 𝐷 ∈ ℕ₀
12		dpmul.e	. . . . . . . 8 ⊢ 𝐸 ∈ ℕ₀
13	12	nn0rei 12413	. . . . . . 7 ⊢ 𝐸 ∈ ℝ
14		dpadd3.f	. . . . . . . 8 ⊢ 𝐹 ∈ ℕ₀
15	14	nn0rei 12413	. . . . . . 7 ⊢ 𝐹 ∈ ℝ
16		dp2cl 32833	. . . . . . 7 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ)
17	13, 15, 16	mp2an 692	. . . . . 6 ⊢ _𝐸𝐹 ∈ ℝ
18		dpcl 32844	. . . . . 6 ⊢ ((𝐷 ∈ ℕ₀ ∧ _𝐸𝐹 ∈ ℝ) → (𝐷._𝐸𝐹) ∈ ℝ)
19	11, 17, 18	mp2an 692	. . . . 5 ⊢ (𝐷._𝐸𝐹) ∈ ℝ
20	19	recni 11148	. . . 4 ⊢ (𝐷._𝐸𝐹) ∈ ℂ
21	10, 20	addcli 11140	. . 3 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ
22		dpmul.g	. . . . 5 ⊢ 𝐺 ∈ ℕ₀
23		dpadd3.h	. . . . . . 7 ⊢ 𝐻 ∈ ℕ₀
24	23	nn0rei 12413	. . . . . 6 ⊢ 𝐻 ∈ ℝ
25		dpadd3.i	. . . . . . 7 ⊢ 𝐼 ∈ ℕ₀
26	25	nn0rei 12413	. . . . . 6 ⊢ 𝐼 ∈ ℝ
27		dp2cl 32833	. . . . . 6 ⊢ ((𝐻 ∈ ℝ ∧ 𝐼 ∈ ℝ) → _𝐻𝐼 ∈ ℝ)
28	24, 26, 27	mp2an 692	. . . . 5 ⊢ _𝐻𝐼 ∈ ℝ
29		dpcl 32844	. . . . 5 ⊢ ((𝐺 ∈ ℕ₀ ∧ _𝐻𝐼 ∈ ℝ) → (𝐺._𝐻𝐼) ∈ ℝ)
30	22, 28, 29	mp2an 692	. . . 4 ⊢ (𝐺._𝐻𝐼) ∈ ℝ
31	30	recni 11148	. . 3 ⊢ (𝐺._𝐻𝐼) ∈ ℂ
32		10nn 12625	. . . . . 6 ⊢ ;10 ∈ ℕ
33	32	decnncl2 12633	. . . . 5 ⊢ ;;100 ∈ ℕ
34	33	nncni 12156	. . . 4 ⊢ ;;100 ∈ ℂ
35	33	nnne0i 12186	. . . 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 11147	. . 3 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100))
39		dpadd3.1	. . . 4 ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼
40	1, 2, 5	dpmul100 32850	. . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶
41	11, 12, 15	dpmul100 32850	. . . . 5 ⊢ ((𝐷._𝐸𝐹) · ;;100) = ;;𝐷𝐸𝐹
42	40, 41	oveq12i 7365	. . . 4 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹)
43	22, 23, 26	dpmul100 32850	. . . 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 11776	. . 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 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 ℕ₀cn0 12402 cdc 12609 cdp2 32824 .cdp 32841

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 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-dec 12610 df-dp2 32825 df-dp 32842

This theorem is referenced by: 1mhdrd 32869 hgt750lem2 34622

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