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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpadd2 | Structured version Visualization version GIF version |
Description: Addition with one decimal, no carry. (Contributed by Thierry Arnoux, 29-Dec-2021.) |
Ref | Expression |
---|---|
dpadd2.a | ⊢ 𝐴 ∈ ℕ0 |
dpadd2.b | ⊢ 𝐵 ∈ ℝ+ |
dpadd2.c | ⊢ 𝐶 ∈ ℕ0 |
dpadd2.d | ⊢ 𝐷 ∈ ℝ+ |
dpadd2.e | ⊢ 𝐸 ∈ ℕ0 |
dpadd2.f | ⊢ 𝐹 ∈ ℝ+ |
dpadd2.g | ⊢ 𝐺 ∈ ℕ0 |
dpadd2.h | ⊢ 𝐻 ∈ ℕ0 |
dpadd2.i | ⊢ (𝐺 + 𝐻) = 𝐼 |
dpadd2.1 | ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹) |
Ref | Expression |
---|---|
dpadd2 | ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpadd2.g | . . . 4 ⊢ 𝐺 ∈ ℕ0 | |
2 | dpadd2.a | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
3 | 2 | nn0rei 12479 | . . . . 5 ⊢ 𝐴 ∈ ℝ |
4 | dpadd2.b | . . . . . 6 ⊢ 𝐵 ∈ ℝ+ | |
5 | rpre 12978 | . . . . . 6 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ 𝐵 ∈ ℝ |
7 | dp2cl 32024 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → _𝐴𝐵 ∈ ℝ) | |
8 | 3, 6, 7 | mp2an 691 | . . . 4 ⊢ _𝐴𝐵 ∈ ℝ |
9 | 1, 8 | dpval2 32037 | . . 3 ⊢ (𝐺._𝐴𝐵) = (𝐺 + (_𝐴𝐵 / ;10)) |
10 | dpadd2.h | . . . 4 ⊢ 𝐻 ∈ ℕ0 | |
11 | dpadd2.c | . . . . . 6 ⊢ 𝐶 ∈ ℕ0 | |
12 | 11 | nn0rei 12479 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
13 | dpadd2.d | . . . . . 6 ⊢ 𝐷 ∈ ℝ+ | |
14 | rpre 12978 | . . . . . 6 ⊢ (𝐷 ∈ ℝ+ → 𝐷 ∈ ℝ) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ 𝐷 ∈ ℝ |
16 | dp2cl 32024 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ) | |
17 | 12, 15, 16 | mp2an 691 | . . . 4 ⊢ _𝐶𝐷 ∈ ℝ |
18 | 10, 17 | dpval2 32037 | . . 3 ⊢ (𝐻._𝐶𝐷) = (𝐻 + (_𝐶𝐷 / ;10)) |
19 | 9, 18 | oveq12i 7416 | . 2 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = ((𝐺 + (_𝐴𝐵 / ;10)) + (𝐻 + (_𝐶𝐷 / ;10))) |
20 | 1 | nn0cni 12480 | . . 3 ⊢ 𝐺 ∈ ℂ |
21 | 8 | recni 11224 | . . . 4 ⊢ _𝐴𝐵 ∈ ℂ |
22 | 10nn 12689 | . . . . 5 ⊢ ;10 ∈ ℕ | |
23 | 22 | nncni 12218 | . . . 4 ⊢ ;10 ∈ ℂ |
24 | 22 | nnne0i 12248 | . . . 4 ⊢ ;10 ≠ 0 |
25 | 21, 23, 24 | divcli 11952 | . . 3 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ |
26 | 10 | nn0cni 12480 | . . 3 ⊢ 𝐻 ∈ ℂ |
27 | 17 | recni 11224 | . . . 4 ⊢ _𝐶𝐷 ∈ ℂ |
28 | 27, 23, 24 | divcli 11952 | . . 3 ⊢ (_𝐶𝐷 / ;10) ∈ ℂ |
29 | 20, 25, 26, 28 | add4i 11434 | . 2 ⊢ ((𝐺 + (_𝐴𝐵 / ;10)) + (𝐻 + (_𝐶𝐷 / ;10))) = ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) |
30 | dpadd2.i | . . . 4 ⊢ (𝐺 + 𝐻) = 𝐼 | |
31 | 21, 27, 23, 24 | divdiri 11967 | . . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10)) |
32 | dpadd2.1 | . . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹) | |
33 | dpval 32034 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | |
34 | 2, 6, 33 | mp2an 691 | . . . . . . . 8 ⊢ (𝐴.𝐵) = _𝐴𝐵 |
35 | dpval 32034 | . . . . . . . . 9 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷) | |
36 | 11, 15, 35 | mp2an 691 | . . . . . . . 8 ⊢ (𝐶.𝐷) = _𝐶𝐷 |
37 | 34, 36 | oveq12i 7416 | . . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (_𝐴𝐵 + _𝐶𝐷) |
38 | dpadd2.e | . . . . . . . 8 ⊢ 𝐸 ∈ ℕ0 | |
39 | dpadd2.f | . . . . . . . . 9 ⊢ 𝐹 ∈ ℝ+ | |
40 | rpre 12978 | . . . . . . . . 9 ⊢ (𝐹 ∈ ℝ+ → 𝐹 ∈ ℝ) | |
41 | 39, 40 | ax-mp 5 | . . . . . . . 8 ⊢ 𝐹 ∈ ℝ |
42 | dpval 32034 | . . . . . . . 8 ⊢ ((𝐸 ∈ ℕ0 ∧ 𝐹 ∈ ℝ) → (𝐸.𝐹) = _𝐸𝐹) | |
43 | 38, 41, 42 | mp2an 691 | . . . . . . 7 ⊢ (𝐸.𝐹) = _𝐸𝐹 |
44 | 32, 37, 43 | 3eqtr3i 2769 | . . . . . 6 ⊢ (_𝐴𝐵 + _𝐶𝐷) = _𝐸𝐹 |
45 | 44 | oveq1i 7414 | . . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = (_𝐸𝐹 / ;10) |
46 | 31, 45 | eqtr3i 2763 | . . . 4 ⊢ ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10)) = (_𝐸𝐹 / ;10) |
47 | 30, 46 | oveq12i 7416 | . . 3 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼 + (_𝐸𝐹 / ;10)) |
48 | 1, 10 | nn0addcli 12505 | . . . . 5 ⊢ (𝐺 + 𝐻) ∈ ℕ0 |
49 | 30, 48 | eqeltrri 2831 | . . . 4 ⊢ 𝐼 ∈ ℕ0 |
50 | 38 | nn0rei 12479 | . . . . 5 ⊢ 𝐸 ∈ ℝ |
51 | dp2cl 32024 | . . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ) | |
52 | 50, 41, 51 | mp2an 691 | . . . 4 ⊢ _𝐸𝐹 ∈ ℝ |
53 | 49, 52 | dpval2 32037 | . . 3 ⊢ (𝐼._𝐸𝐹) = (𝐼 + (_𝐸𝐹 / ;10)) |
54 | 47, 53 | eqtr4i 2764 | . 2 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼._𝐸𝐹) |
55 | 19, 29, 54 | 3eqtri 2765 | 1 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7404 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 / cdiv 11867 ℕ0cn0 12468 ;cdc 12673 ℝ+crp 12970 _cdp2 32015 .cdp 32032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-dec 12674 df-rp 12971 df-dp2 32016 df-dp 32033 |
This theorem is referenced by: hgt750lemd 33598 |
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