Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpadd3 | Structured version Visualization version GIF version |
Description: Addition with two decimals. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
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 |
Ref | Expression |
---|---|
dpadd3 | ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpmul.a | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
2 | dpmul.b | . . . . . . . 8 ⊢ 𝐵 ∈ ℕ0 | |
3 | 2 | nn0rei 11896 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
4 | dpmul.c | . . . . . . . 8 ⊢ 𝐶 ∈ ℕ0 | |
5 | 4 | nn0rei 11896 | . . . . . . 7 ⊢ 𝐶 ∈ ℝ |
6 | dp2cl 30483 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ) | |
7 | 3, 5, 6 | mp2an 688 | . . . . . 6 ⊢ _𝐵𝐶 ∈ ℝ |
8 | dpcl 30494 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ _𝐵𝐶 ∈ ℝ) → (𝐴._𝐵𝐶) ∈ ℝ) | |
9 | 1, 7, 8 | mp2an 688 | . . . . 5 ⊢ (𝐴._𝐵𝐶) ∈ ℝ |
10 | 9 | recni 10643 | . . . 4 ⊢ (𝐴._𝐵𝐶) ∈ ℂ |
11 | dpmul.d | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
12 | dpmul.e | . . . . . . . 8 ⊢ 𝐸 ∈ ℕ0 | |
13 | 12 | nn0rei 11896 | . . . . . . 7 ⊢ 𝐸 ∈ ℝ |
14 | dpadd3.f | . . . . . . . 8 ⊢ 𝐹 ∈ ℕ0 | |
15 | 14 | nn0rei 11896 | . . . . . . 7 ⊢ 𝐹 ∈ ℝ |
16 | dp2cl 30483 | . . . . . . 7 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ) | |
17 | 13, 15, 16 | mp2an 688 | . . . . . 6 ⊢ _𝐸𝐹 ∈ ℝ |
18 | dpcl 30494 | . . . . . 6 ⊢ ((𝐷 ∈ ℕ0 ∧ _𝐸𝐹 ∈ ℝ) → (𝐷._𝐸𝐹) ∈ ℝ) | |
19 | 11, 17, 18 | mp2an 688 | . . . . 5 ⊢ (𝐷._𝐸𝐹) ∈ ℝ |
20 | 19 | recni 10643 | . . . 4 ⊢ (𝐷._𝐸𝐹) ∈ ℂ |
21 | 10, 20 | addcli 10635 | . . 3 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ |
22 | dpmul.g | . . . . 5 ⊢ 𝐺 ∈ ℕ0 | |
23 | dpadd3.h | . . . . . . 7 ⊢ 𝐻 ∈ ℕ0 | |
24 | 23 | nn0rei 11896 | . . . . . 6 ⊢ 𝐻 ∈ ℝ |
25 | dpadd3.i | . . . . . . 7 ⊢ 𝐼 ∈ ℕ0 | |
26 | 25 | nn0rei 11896 | . . . . . 6 ⊢ 𝐼 ∈ ℝ |
27 | dp2cl 30483 | . . . . . 6 ⊢ ((𝐻 ∈ ℝ ∧ 𝐼 ∈ ℝ) → _𝐻𝐼 ∈ ℝ) | |
28 | 24, 26, 27 | mp2an 688 | . . . . 5 ⊢ _𝐻𝐼 ∈ ℝ |
29 | dpcl 30494 | . . . . 5 ⊢ ((𝐺 ∈ ℕ0 ∧ _𝐻𝐼 ∈ ℝ) → (𝐺._𝐻𝐼) ∈ ℝ) | |
30 | 22, 28, 29 | mp2an 688 | . . . 4 ⊢ (𝐺._𝐻𝐼) ∈ ℝ |
31 | 30 | recni 10643 | . . 3 ⊢ (𝐺._𝐻𝐼) ∈ ℂ |
32 | 10nn 12102 | . . . . . 6 ⊢ ;10 ∈ ℕ | |
33 | 32 | decnncl2 12110 | . . . . 5 ⊢ ;;100 ∈ ℕ |
34 | 33 | nncni 11636 | . . . 4 ⊢ ;;100 ∈ ℂ |
35 | 33 | nnne0i 11665 | . . . 4 ⊢ ;;100 ≠ 0 |
36 | 34, 35 | pm3.2i 471 | . . 3 ⊢ (;;100 ∈ ℂ ∧ ;;100 ≠ 0) |
37 | 21, 31, 36 | 3pm3.2i 1331 | . 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ ∧ (𝐺._𝐻𝐼) ∈ ℂ ∧ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)) |
38 | 10, 20, 34 | adddiri 10642 | . . 3 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) |
39 | dpadd3.1 | . . . 4 ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼 | |
40 | 1, 2, 5 | dpmul100 30500 | . . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶 |
41 | 11, 12, 15 | dpmul100 30500 | . . . . 5 ⊢ ((𝐷._𝐸𝐹) · ;;100) = ;;𝐷𝐸𝐹 |
42 | 40, 41 | oveq12i 7157 | . . . 4 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) |
43 | 22, 23, 26 | dpmul100 30500 | . . . 4 ⊢ ((𝐺._𝐻𝐼) · ;;100) = ;;𝐺𝐻𝐼 |
44 | 39, 42, 43 | 3eqtr4i 2851 | . . 3 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = ((𝐺._𝐻𝐼) · ;;100) |
45 | 38, 44 | eqtri 2841 | . 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = ((𝐺._𝐻𝐼) · ;;100) |
46 | mulcan2 11266 | . . 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 477 | . 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 688 | 1 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 ℕ0cn0 11885 ;cdc 12086 _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-dp2 30475 df-dp 30492 |
This theorem is referenced by: 1mhdrd 30519 hgt750lem2 31822 |
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