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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 12479 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
4 | dpmul.c | . . . . . . . 8 ⊢ 𝐶 ∈ ℕ0 | |
5 | 4 | nn0rei 12479 | . . . . . . 7 ⊢ 𝐶 ∈ ℝ |
6 | dp2cl 32479 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ) | |
7 | 3, 5, 6 | mp2an 689 | . . . . . 6 ⊢ _𝐵𝐶 ∈ ℝ |
8 | dpcl 32490 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ _𝐵𝐶 ∈ ℝ) → (𝐴._𝐵𝐶) ∈ ℝ) | |
9 | 1, 7, 8 | mp2an 689 | . . . . 5 ⊢ (𝐴._𝐵𝐶) ∈ ℝ |
10 | 9 | recni 11224 | . . . 4 ⊢ (𝐴._𝐵𝐶) ∈ ℂ |
11 | dpmul.d | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
12 | dpmul.e | . . . . . . . 8 ⊢ 𝐸 ∈ ℕ0 | |
13 | 12 | nn0rei 12479 | . . . . . . 7 ⊢ 𝐸 ∈ ℝ |
14 | dpadd3.f | . . . . . . . 8 ⊢ 𝐹 ∈ ℕ0 | |
15 | 14 | nn0rei 12479 | . . . . . . 7 ⊢ 𝐹 ∈ ℝ |
16 | dp2cl 32479 | . . . . . . 7 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ) | |
17 | 13, 15, 16 | mp2an 689 | . . . . . 6 ⊢ _𝐸𝐹 ∈ ℝ |
18 | dpcl 32490 | . . . . . 6 ⊢ ((𝐷 ∈ ℕ0 ∧ _𝐸𝐹 ∈ ℝ) → (𝐷._𝐸𝐹) ∈ ℝ) | |
19 | 11, 17, 18 | mp2an 689 | . . . . 5 ⊢ (𝐷._𝐸𝐹) ∈ ℝ |
20 | 19 | recni 11224 | . . . 4 ⊢ (𝐷._𝐸𝐹) ∈ ℂ |
21 | 10, 20 | addcli 11216 | . . 3 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ |
22 | dpmul.g | . . . . 5 ⊢ 𝐺 ∈ ℕ0 | |
23 | dpadd3.h | . . . . . . 7 ⊢ 𝐻 ∈ ℕ0 | |
24 | 23 | nn0rei 12479 | . . . . . 6 ⊢ 𝐻 ∈ ℝ |
25 | dpadd3.i | . . . . . . 7 ⊢ 𝐼 ∈ ℕ0 | |
26 | 25 | nn0rei 12479 | . . . . . 6 ⊢ 𝐼 ∈ ℝ |
27 | dp2cl 32479 | . . . . . 6 ⊢ ((𝐻 ∈ ℝ ∧ 𝐼 ∈ ℝ) → _𝐻𝐼 ∈ ℝ) | |
28 | 24, 26, 27 | mp2an 689 | . . . . 5 ⊢ _𝐻𝐼 ∈ ℝ |
29 | dpcl 32490 | . . . . 5 ⊢ ((𝐺 ∈ ℕ0 ∧ _𝐻𝐼 ∈ ℝ) → (𝐺._𝐻𝐼) ∈ ℝ) | |
30 | 22, 28, 29 | mp2an 689 | . . . 4 ⊢ (𝐺._𝐻𝐼) ∈ ℝ |
31 | 30 | recni 11224 | . . 3 ⊢ (𝐺._𝐻𝐼) ∈ ℂ |
32 | 10nn 12689 | . . . . . 6 ⊢ ;10 ∈ ℕ | |
33 | 32 | decnncl2 12697 | . . . . 5 ⊢ ;;100 ∈ ℕ |
34 | 33 | nncni 12218 | . . . 4 ⊢ ;;100 ∈ ℂ |
35 | 33 | nnne0i 12248 | . . . 4 ⊢ ;;100 ≠ 0 |
36 | 34, 35 | pm3.2i 470 | . . 3 ⊢ (;;100 ∈ ℂ ∧ ;;100 ≠ 0) |
37 | 21, 31, 36 | 3pm3.2i 1336 | . 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ ∧ (𝐺._𝐻𝐼) ∈ ℂ ∧ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)) |
38 | 10, 20, 34 | adddiri 11223 | . . 3 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) |
39 | dpadd3.1 | . . . 4 ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼 | |
40 | 1, 2, 5 | dpmul100 32496 | . . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶 |
41 | 11, 12, 15 | dpmul100 32496 | . . . . 5 ⊢ ((𝐷._𝐸𝐹) · ;;100) = ;;𝐷𝐸𝐹 |
42 | 40, 41 | oveq12i 7413 | . . . 4 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) |
43 | 22, 23, 26 | dpmul100 32496 | . . . 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 11848 | . . 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 689 | 1 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 (class class class)co 7401 ℂcc 11103 ℝcr 11104 0cc0 11105 1c1 11106 + caddc 11108 · cmul 11110 ℕ0cn0 12468 ;cdc 12673 _cdp2 32470 .cdp 32487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 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-dp2 32471 df-dp 32488 |
This theorem is referenced by: 1mhdrd 32515 hgt750lem2 34119 |
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