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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 11762 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
4 | dpmul.c | . . . . . . . 8 ⊢ 𝐶 ∈ ℕ0 | |
5 | 4 | nn0rei 11762 | . . . . . . 7 ⊢ 𝐶 ∈ ℝ |
6 | dp2cl 30236 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ) | |
7 | 3, 5, 6 | mp2an 688 | . . . . . 6 ⊢ _𝐵𝐶 ∈ ℝ |
8 | dpcl 30247 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ _𝐵𝐶 ∈ ℝ) → (𝐴._𝐵𝐶) ∈ ℝ) | |
9 | 1, 7, 8 | mp2an 688 | . . . . 5 ⊢ (𝐴._𝐵𝐶) ∈ ℝ |
10 | 9 | recni 10508 | . . . 4 ⊢ (𝐴._𝐵𝐶) ∈ ℂ |
11 | dpmul.d | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
12 | dpmul.e | . . . . . . . 8 ⊢ 𝐸 ∈ ℕ0 | |
13 | 12 | nn0rei 11762 | . . . . . . 7 ⊢ 𝐸 ∈ ℝ |
14 | dpadd3.f | . . . . . . . 8 ⊢ 𝐹 ∈ ℕ0 | |
15 | 14 | nn0rei 11762 | . . . . . . 7 ⊢ 𝐹 ∈ ℝ |
16 | dp2cl 30236 | . . . . . . 7 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ) | |
17 | 13, 15, 16 | mp2an 688 | . . . . . 6 ⊢ _𝐸𝐹 ∈ ℝ |
18 | dpcl 30247 | . . . . . 6 ⊢ ((𝐷 ∈ ℕ0 ∧ _𝐸𝐹 ∈ ℝ) → (𝐷._𝐸𝐹) ∈ ℝ) | |
19 | 11, 17, 18 | mp2an 688 | . . . . 5 ⊢ (𝐷._𝐸𝐹) ∈ ℝ |
20 | 19 | recni 10508 | . . . 4 ⊢ (𝐷._𝐸𝐹) ∈ ℂ |
21 | 10, 20 | addcli 10500 | . . 3 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ |
22 | dpmul.g | . . . . 5 ⊢ 𝐺 ∈ ℕ0 | |
23 | dpadd3.h | . . . . . . 7 ⊢ 𝐻 ∈ ℕ0 | |
24 | 23 | nn0rei 11762 | . . . . . 6 ⊢ 𝐻 ∈ ℝ |
25 | dpadd3.i | . . . . . . 7 ⊢ 𝐼 ∈ ℕ0 | |
26 | 25 | nn0rei 11762 | . . . . . 6 ⊢ 𝐼 ∈ ℝ |
27 | dp2cl 30236 | . . . . . 6 ⊢ ((𝐻 ∈ ℝ ∧ 𝐼 ∈ ℝ) → _𝐻𝐼 ∈ ℝ) | |
28 | 24, 26, 27 | mp2an 688 | . . . . 5 ⊢ _𝐻𝐼 ∈ ℝ |
29 | dpcl 30247 | . . . . 5 ⊢ ((𝐺 ∈ ℕ0 ∧ _𝐻𝐼 ∈ ℝ) → (𝐺._𝐻𝐼) ∈ ℝ) | |
30 | 22, 28, 29 | mp2an 688 | . . . 4 ⊢ (𝐺._𝐻𝐼) ∈ ℝ |
31 | 30 | recni 10508 | . . 3 ⊢ (𝐺._𝐻𝐼) ∈ ℂ |
32 | 10nn 11968 | . . . . . 6 ⊢ ;10 ∈ ℕ | |
33 | 32 | decnncl2 11976 | . . . . 5 ⊢ ;;100 ∈ ℕ |
34 | 33 | nncni 11502 | . . . 4 ⊢ ;;100 ∈ ℂ |
35 | 33 | nnne0i 11531 | . . . 4 ⊢ ;;100 ≠ 0 |
36 | 34, 35 | pm3.2i 471 | . . 3 ⊢ (;;100 ∈ ℂ ∧ ;;100 ≠ 0) |
37 | 21, 31, 36 | 3pm3.2i 1332 | . 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ ∧ (𝐺._𝐻𝐼) ∈ ℂ ∧ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)) |
38 | 10, 20, 34 | adddiri 10507 | . . 3 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) |
39 | dpadd3.1 | . . . 4 ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼 | |
40 | 1, 2, 5 | dpmul100 30253 | . . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶 |
41 | 11, 12, 15 | dpmul100 30253 | . . . . 5 ⊢ ((𝐷._𝐸𝐹) · ;;100) = ;;𝐷𝐸𝐹 |
42 | 40, 41 | oveq12i 7035 | . . . 4 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) |
43 | 22, 23, 26 | dpmul100 30253 | . . . 4 ⊢ ((𝐺._𝐻𝐼) · ;;100) = ;;𝐺𝐻𝐼 |
44 | 39, 42, 43 | 3eqtr4i 2831 | . . 3 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = ((𝐺._𝐻𝐼) · ;;100) |
45 | 38, 44 | eqtri 2821 | . 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = ((𝐺._𝐻𝐼) · ;;100) |
46 | mulcan2 11132 | . . 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 1080 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 (class class class)co 7023 ℂcc 10388 ℝcr 10389 0cc0 10390 1c1 10391 + caddc 10393 · cmul 10395 ℕ0cn0 11751 ;cdc 11952 _cdp2 30227 .cdp 30244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-dec 11953 df-dp2 30228 df-dp 30245 |
This theorem is referenced by: 1mhdrd 30272 hgt750lem2 31536 |
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