|   | Mathbox for Thierry Arnoux | < Previous  
      Next > Nearby theorems | |
| 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 12539 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ | 
| 4 | dpmul.c | . . . . . . . 8 ⊢ 𝐶 ∈ ℕ0 | |
| 5 | 4 | nn0rei 12539 | . . . . . . 7 ⊢ 𝐶 ∈ ℝ | 
| 6 | dp2cl 32863 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ) | |
| 7 | 3, 5, 6 | mp2an 692 | . . . . . 6 ⊢ _𝐵𝐶 ∈ ℝ | 
| 8 | dpcl 32874 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ _𝐵𝐶 ∈ ℝ) → (𝐴._𝐵𝐶) ∈ ℝ) | |
| 9 | 1, 7, 8 | mp2an 692 | . . . . 5 ⊢ (𝐴._𝐵𝐶) ∈ ℝ | 
| 10 | 9 | recni 11276 | . . . 4 ⊢ (𝐴._𝐵𝐶) ∈ ℂ | 
| 11 | dpmul.d | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
| 12 | dpmul.e | . . . . . . . 8 ⊢ 𝐸 ∈ ℕ0 | |
| 13 | 12 | nn0rei 12539 | . . . . . . 7 ⊢ 𝐸 ∈ ℝ | 
| 14 | dpadd3.f | . . . . . . . 8 ⊢ 𝐹 ∈ ℕ0 | |
| 15 | 14 | nn0rei 12539 | . . . . . . 7 ⊢ 𝐹 ∈ ℝ | 
| 16 | dp2cl 32863 | . . . . . . 7 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ) | |
| 17 | 13, 15, 16 | mp2an 692 | . . . . . 6 ⊢ _𝐸𝐹 ∈ ℝ | 
| 18 | dpcl 32874 | . . . . . 6 ⊢ ((𝐷 ∈ ℕ0 ∧ _𝐸𝐹 ∈ ℝ) → (𝐷._𝐸𝐹) ∈ ℝ) | |
| 19 | 11, 17, 18 | mp2an 692 | . . . . 5 ⊢ (𝐷._𝐸𝐹) ∈ ℝ | 
| 20 | 19 | recni 11276 | . . . 4 ⊢ (𝐷._𝐸𝐹) ∈ ℂ | 
| 21 | 10, 20 | addcli 11268 | . . 3 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ | 
| 22 | dpmul.g | . . . . 5 ⊢ 𝐺 ∈ ℕ0 | |
| 23 | dpadd3.h | . . . . . . 7 ⊢ 𝐻 ∈ ℕ0 | |
| 24 | 23 | nn0rei 12539 | . . . . . 6 ⊢ 𝐻 ∈ ℝ | 
| 25 | dpadd3.i | . . . . . . 7 ⊢ 𝐼 ∈ ℕ0 | |
| 26 | 25 | nn0rei 12539 | . . . . . 6 ⊢ 𝐼 ∈ ℝ | 
| 27 | dp2cl 32863 | . . . . . 6 ⊢ ((𝐻 ∈ ℝ ∧ 𝐼 ∈ ℝ) → _𝐻𝐼 ∈ ℝ) | |
| 28 | 24, 26, 27 | mp2an 692 | . . . . 5 ⊢ _𝐻𝐼 ∈ ℝ | 
| 29 | dpcl 32874 | . . . . 5 ⊢ ((𝐺 ∈ ℕ0 ∧ _𝐻𝐼 ∈ ℝ) → (𝐺._𝐻𝐼) ∈ ℝ) | |
| 30 | 22, 28, 29 | mp2an 692 | . . . 4 ⊢ (𝐺._𝐻𝐼) ∈ ℝ | 
| 31 | 30 | recni 11276 | . . 3 ⊢ (𝐺._𝐻𝐼) ∈ ℂ | 
| 32 | 10nn 12751 | . . . . . 6 ⊢ ;10 ∈ ℕ | |
| 33 | 32 | decnncl2 12759 | . . . . 5 ⊢ ;;100 ∈ ℕ | 
| 34 | 33 | nncni 12277 | . . . 4 ⊢ ;;100 ∈ ℂ | 
| 35 | 33 | nnne0i 12307 | . . . 4 ⊢ ;;100 ≠ 0 | 
| 36 | 34, 35 | pm3.2i 470 | . . 3 ⊢ (;;100 ∈ ℂ ∧ ;;100 ≠ 0) | 
| 37 | 21, 31, 36 | 3pm3.2i 1339 | . 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ ∧ (𝐺._𝐻𝐼) ∈ ℂ ∧ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)) | 
| 38 | 10, 20, 34 | adddiri 11275 | . . 3 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) | 
| 39 | dpadd3.1 | . . . 4 ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼 | |
| 40 | 1, 2, 5 | dpmul100 32880 | . . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶 | 
| 41 | 11, 12, 15 | dpmul100 32880 | . . . . 5 ⊢ ((𝐷._𝐸𝐹) · ;;100) = ;;𝐷𝐸𝐹 | 
| 42 | 40, 41 | oveq12i 7444 | . . . 4 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) | 
| 43 | 22, 23, 26 | dpmul100 32880 | . . . 4 ⊢ ((𝐺._𝐻𝐼) · ;;100) = ;;𝐺𝐻𝐼 | 
| 44 | 39, 42, 43 | 3eqtr4i 2774 | . . 3 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = ((𝐺._𝐻𝐼) · ;;100) | 
| 45 | 38, 44 | eqtri 2764 | . 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = ((𝐺._𝐻𝐼) · ;;100) | 
| 46 | mulcan2 11902 | . . 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 692 | 1 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 ℕ0cn0 12528 ;cdc 12735 _cdp2 32854 .cdp 32871 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-dec 12736 df-dp2 32855 df-dp 32872 | 
| This theorem is referenced by: 1mhdrd 32899 hgt750lem2 34668 | 
| Copyright terms: Public domain | W3C validator |