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 11303 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
4 | dpmul.c | . . . . . . . 8 ⊢ 𝐶 ∈ ℕ0 | |
5 | 4 | nn0rei 11303 | . . . . . . 7 ⊢ 𝐶 ∈ ℝ |
6 | dp2cl 29587 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → _𝐵𝐶 ∈ ℝ) | |
7 | 3, 5, 6 | mp2an 708 | . . . . . 6 ⊢ _𝐵𝐶 ∈ ℝ |
8 | dpcl 29598 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ _𝐵𝐶 ∈ ℝ) → (𝐴._𝐵𝐶) ∈ ℝ) | |
9 | 1, 7, 8 | mp2an 708 | . . . . 5 ⊢ (𝐴._𝐵𝐶) ∈ ℝ |
10 | 9 | recni 10052 | . . . 4 ⊢ (𝐴._𝐵𝐶) ∈ ℂ |
11 | dpmul.d | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
12 | dpmul.e | . . . . . . . 8 ⊢ 𝐸 ∈ ℕ0 | |
13 | 12 | nn0rei 11303 | . . . . . . 7 ⊢ 𝐸 ∈ ℝ |
14 | dpadd3.f | . . . . . . . 8 ⊢ 𝐹 ∈ ℕ0 | |
15 | 14 | nn0rei 11303 | . . . . . . 7 ⊢ 𝐹 ∈ ℝ |
16 | dp2cl 29587 | . . . . . . 7 ⊢ ((𝐸 ∈ ℝ ∧ 𝐹 ∈ ℝ) → _𝐸𝐹 ∈ ℝ) | |
17 | 13, 15, 16 | mp2an 708 | . . . . . 6 ⊢ _𝐸𝐹 ∈ ℝ |
18 | dpcl 29598 | . . . . . 6 ⊢ ((𝐷 ∈ ℕ0 ∧ _𝐸𝐹 ∈ ℝ) → (𝐷._𝐸𝐹) ∈ ℝ) | |
19 | 11, 17, 18 | mp2an 708 | . . . . 5 ⊢ (𝐷._𝐸𝐹) ∈ ℝ |
20 | 19 | recni 10052 | . . . 4 ⊢ (𝐷._𝐸𝐹) ∈ ℂ |
21 | 10, 20 | addcli 10044 | . . 3 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ |
22 | dpmul.g | . . . . 5 ⊢ 𝐺 ∈ ℕ0 | |
23 | dpadd3.h | . . . . . . 7 ⊢ 𝐻 ∈ ℕ0 | |
24 | 23 | nn0rei 11303 | . . . . . 6 ⊢ 𝐻 ∈ ℝ |
25 | dpadd3.i | . . . . . . 7 ⊢ 𝐼 ∈ ℕ0 | |
26 | 25 | nn0rei 11303 | . . . . . 6 ⊢ 𝐼 ∈ ℝ |
27 | dp2cl 29587 | . . . . . 6 ⊢ ((𝐻 ∈ ℝ ∧ 𝐼 ∈ ℝ) → _𝐻𝐼 ∈ ℝ) | |
28 | 24, 26, 27 | mp2an 708 | . . . . 5 ⊢ _𝐻𝐼 ∈ ℝ |
29 | dpcl 29598 | . . . . 5 ⊢ ((𝐺 ∈ ℕ0 ∧ _𝐻𝐼 ∈ ℝ) → (𝐺._𝐻𝐼) ∈ ℝ) | |
30 | 22, 28, 29 | mp2an 708 | . . . 4 ⊢ (𝐺._𝐻𝐼) ∈ ℝ |
31 | 30 | recni 10052 | . . 3 ⊢ (𝐺._𝐻𝐼) ∈ ℂ |
32 | 10nn 11514 | . . . . . 6 ⊢ ;10 ∈ ℕ | |
33 | 32 | decnncl2 11525 | . . . . 5 ⊢ ;;100 ∈ ℕ |
34 | 33 | nncni 11030 | . . . 4 ⊢ ;;100 ∈ ℂ |
35 | 33 | nnne0i 11055 | . . . 4 ⊢ ;;100 ≠ 0 |
36 | 34, 35 | pm3.2i 471 | . . 3 ⊢ (;;100 ∈ ℂ ∧ ;;100 ≠ 0) |
37 | 21, 31, 36 | 3pm3.2i 1239 | . 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) ∈ ℂ ∧ (𝐺._𝐻𝐼) ∈ ℂ ∧ (;;100 ∈ ℂ ∧ ;;100 ≠ 0)) |
38 | 10, 20, 34 | adddiri 10051 | . . 3 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) |
39 | dpadd3.1 | . . . 4 ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼 | |
40 | 1, 2, 5 | dpmul100 29605 | . . . . 5 ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶 |
41 | 11, 12, 15 | dpmul100 29605 | . . . . 5 ⊢ ((𝐷._𝐸𝐹) · ;;100) = ;;𝐷𝐸𝐹 |
42 | 40, 41 | oveq12i 6662 | . . . 4 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) |
43 | 22, 23, 26 | dpmul100 29605 | . . . 4 ⊢ ((𝐺._𝐻𝐼) · ;;100) = ;;𝐺𝐻𝐼 |
44 | 39, 42, 43 | 3eqtr4i 2654 | . . 3 ⊢ (((𝐴._𝐵𝐶) · ;;100) + ((𝐷._𝐸𝐹) · ;;100)) = ((𝐺._𝐻𝐼) · ;;100) |
45 | 38, 44 | eqtri 2644 | . 2 ⊢ (((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) · ;;100) = ((𝐺._𝐻𝐼) · ;;100) |
46 | mulcan2 10665 | . . 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 501 | . 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 708 | 1 ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℕ0cn0 11292 ;cdc 11493 _cdp2 29577 .cdp 29595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-dec 11494 df-dp2 29578 df-dp 29596 |
This theorem is referenced by: 1mhdrd 29624 hgt750lem2 30730 |
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