Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 11896 | . . . . 5 ⊢ 𝐴 ∈ ℝ |
4 | dpadd2.b | . . . . . 6 ⊢ 𝐵 ∈ ℝ+ | |
5 | rpre 12385 | . . . . . 6 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ 𝐵 ∈ ℝ |
7 | dp2cl 30483 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → _𝐴𝐵 ∈ ℝ) | |
8 | 3, 6, 7 | mp2an 688 | . . . 4 ⊢ _𝐴𝐵 ∈ ℝ |
9 | 1, 8 | dpval2 30496 | . . 3 ⊢ (𝐺._𝐴𝐵) = (𝐺 + (_𝐴𝐵 / ;10)) |
10 | dpadd2.h | . . . 4 ⊢ 𝐻 ∈ ℕ0 | |
11 | dpadd2.c | . . . . . 6 ⊢ 𝐶 ∈ ℕ0 | |
12 | 11 | nn0rei 11896 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
13 | dpadd2.d | . . . . . 6 ⊢ 𝐷 ∈ ℝ+ | |
14 | rpre 12385 | . . . . . 6 ⊢ (𝐷 ∈ ℝ+ → 𝐷 ∈ ℝ) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ 𝐷 ∈ ℝ |
16 | dp2cl 30483 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → _𝐶𝐷 ∈ ℝ) | |
17 | 12, 15, 16 | mp2an 688 | . . . 4 ⊢ _𝐶𝐷 ∈ ℝ |
18 | 10, 17 | dpval2 30496 | . . 3 ⊢ (𝐻._𝐶𝐷) = (𝐻 + (_𝐶𝐷 / ;10)) |
19 | 9, 18 | oveq12i 7157 | . 2 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = ((𝐺 + (_𝐴𝐵 / ;10)) + (𝐻 + (_𝐶𝐷 / ;10))) |
20 | 1 | nn0cni 11897 | . . 3 ⊢ 𝐺 ∈ ℂ |
21 | 8 | recni 10643 | . . . 4 ⊢ _𝐴𝐵 ∈ ℂ |
22 | 10nn 12102 | . . . . 5 ⊢ ;10 ∈ ℕ | |
23 | 22 | nncni 11636 | . . . 4 ⊢ ;10 ∈ ℂ |
24 | 22 | nnne0i 11665 | . . . 4 ⊢ ;10 ≠ 0 |
25 | 21, 23, 24 | divcli 11370 | . . 3 ⊢ (_𝐴𝐵 / ;10) ∈ ℂ |
26 | 10 | nn0cni 11897 | . . 3 ⊢ 𝐻 ∈ ℂ |
27 | 17 | recni 10643 | . . . 4 ⊢ _𝐶𝐷 ∈ ℂ |
28 | 27, 23, 24 | divcli 11370 | . . 3 ⊢ (_𝐶𝐷 / ;10) ∈ ℂ |
29 | 20, 25, 26, 28 | add4i 10852 | . 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 11385 | . . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10)) |
32 | dpadd2.1 | . . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹) | |
33 | dpval 30493 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ) → (𝐴.𝐵) = _𝐴𝐵) | |
34 | 2, 6, 33 | mp2an 688 | . . . . . . . 8 ⊢ (𝐴.𝐵) = _𝐴𝐵 |
35 | dpval 30493 | . . . . . . . . 9 ⊢ ((𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℝ) → (𝐶.𝐷) = _𝐶𝐷) | |
36 | 11, 15, 35 | mp2an 688 | . . . . . . . 8 ⊢ (𝐶.𝐷) = _𝐶𝐷 |
37 | 34, 36 | oveq12i 7157 | . . . . . . 7 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (_𝐴𝐵 + _𝐶𝐷) |
38 | dpadd2.e | . . . . . . . 8 ⊢ 𝐸 ∈ ℕ0 | |
39 | dpadd2.f | . . . . . . . . 9 ⊢ 𝐹 ∈ ℝ+ | |
40 | rpre 12385 | . . . . . . . . 9 ⊢ (𝐹 ∈ ℝ+ → 𝐹 ∈ ℝ) | |
41 | 39, 40 | ax-mp 5 | . . . . . . . 8 ⊢ 𝐹 ∈ ℝ |
42 | dpval 30493 | . . . . . . . 8 ⊢ ((𝐸 ∈ ℕ0 ∧ 𝐹 ∈ ℝ) → (𝐸.𝐹) = _𝐸𝐹) | |
43 | 38, 41, 42 | mp2an 688 | . . . . . . 7 ⊢ (𝐸.𝐹) = _𝐸𝐹 |
44 | 32, 37, 43 | 3eqtr3i 2849 | . . . . . 6 ⊢ (_𝐴𝐵 + _𝐶𝐷) = _𝐸𝐹 |
45 | 44 | oveq1i 7155 | . . . . 5 ⊢ ((_𝐴𝐵 + _𝐶𝐷) / ;10) = (_𝐸𝐹 / ;10) |
46 | 31, 45 | eqtr3i 2843 | . . . 4 ⊢ ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10)) = (_𝐸𝐹 / ;10) |
47 | 30, 46 | oveq12i 7157 | . . 3 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼 + (_𝐸𝐹 / ;10)) |
48 | 1, 10 | nn0addcli 11922 | . . . . 5 ⊢ (𝐺 + 𝐻) ∈ ℕ0 |
49 | 30, 48 | eqeltrri 2907 | . . . 4 ⊢ 𝐼 ∈ ℕ0 |
50 | 38 | nn0rei 11896 | . . . . 5 ⊢ 𝐸 ∈ ℝ |
51 | dp2cl 30483 | . . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ) | |
52 | 50, 41, 51 | mp2an 688 | . . . 4 ⊢ _𝐸𝐹 ∈ ℝ |
53 | 49, 52 | dpval2 30496 | . . 3 ⊢ (𝐼._𝐸𝐹) = (𝐼 + (_𝐸𝐹 / ;10)) |
54 | 47, 53 | eqtr4i 2844 | . 2 ⊢ ((𝐺 + 𝐻) + ((_𝐴𝐵 / ;10) + (_𝐶𝐷 / ;10))) = (𝐼._𝐸𝐹) |
55 | 19, 29, 54 | 3eqtri 2845 | 1 ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 / cdiv 11285 ℕ0cn0 11885 ;cdc 12086 ℝ+crp 12377 _cdp2 30474 .cdp 30491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-dec 12087 df-rp 12378 df-dp2 30475 df-dp 30492 |
This theorem is referenced by: hgt750lemd 31818 |
Copyright terms: Public domain | W3C validator |