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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpadd | Structured version Visualization version GIF version |
Description: Addition with one decimal. (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 |
dpadd.f | ⊢ 𝐹 ∈ ℕ0 |
dpadd.1 | ⊢ (;𝐴𝐵 + ;𝐶𝐷) = ;𝐸𝐹 |
Ref | Expression |
---|---|
dpadd | ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpmul.a | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
2 | dpmul.b | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
3 | 1, 2 | deccl 12714 | . . . . 5 ⊢ ;𝐴𝐵 ∈ ℕ0 |
4 | 3 | nn0cni 12506 | . . . 4 ⊢ ;𝐴𝐵 ∈ ℂ |
5 | dpmul.c | . . . . . 6 ⊢ 𝐶 ∈ ℕ0 | |
6 | dpmul.d | . . . . . 6 ⊢ 𝐷 ∈ ℕ0 | |
7 | 5, 6 | deccl 12714 | . . . . 5 ⊢ ;𝐶𝐷 ∈ ℕ0 |
8 | 7 | nn0cni 12506 | . . . 4 ⊢ ;𝐶𝐷 ∈ ℂ |
9 | 10nn 12715 | . . . . 5 ⊢ ;10 ∈ ℕ | |
10 | 9 | nncni 12244 | . . . 4 ⊢ ;10 ∈ ℂ |
11 | 9 | nnne0i 12274 | . . . 4 ⊢ ;10 ≠ 0 |
12 | 4, 8, 10, 11 | divdiri 11993 | . . 3 ⊢ ((;𝐴𝐵 + ;𝐶𝐷) / ;10) = ((;𝐴𝐵 / ;10) + (;𝐶𝐷 / ;10)) |
13 | dpadd.1 | . . . 4 ⊢ (;𝐴𝐵 + ;𝐶𝐷) = ;𝐸𝐹 | |
14 | 13 | oveq1i 7424 | . . 3 ⊢ ((;𝐴𝐵 + ;𝐶𝐷) / ;10) = (;𝐸𝐹 / ;10) |
15 | 12, 14 | eqtr3i 2757 | . 2 ⊢ ((;𝐴𝐵 / ;10) + (;𝐶𝐷 / ;10)) = (;𝐸𝐹 / ;10) |
16 | 2 | nn0rei 12505 | . . . 4 ⊢ 𝐵 ∈ ℝ |
17 | 1, 16 | decdiv10 32601 | . . 3 ⊢ (;𝐴𝐵 / ;10) = (𝐴.𝐵) |
18 | 6 | nn0rei 12505 | . . . 4 ⊢ 𝐷 ∈ ℝ |
19 | 5, 18 | decdiv10 32601 | . . 3 ⊢ (;𝐶𝐷 / ;10) = (𝐶.𝐷) |
20 | 17, 19 | oveq12i 7426 | . 2 ⊢ ((;𝐴𝐵 / ;10) + (;𝐶𝐷 / ;10)) = ((𝐴.𝐵) + (𝐶.𝐷)) |
21 | dpmul.e | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
22 | dpadd.f | . . . 4 ⊢ 𝐹 ∈ ℕ0 | |
23 | 22 | nn0rei 12505 | . . 3 ⊢ 𝐹 ∈ ℝ |
24 | 21, 23 | decdiv10 32601 | . 2 ⊢ (;𝐸𝐹 / ;10) = (𝐸.𝐹) |
25 | 15, 20, 24 | 3eqtr3i 2763 | 1 ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7414 0cc0 11130 1c1 11131 + caddc 11133 / cdiv 11893 ℕ0cn0 12494 ;cdc 12699 .cdp 32593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-dec 12700 df-dp2 32577 df-dp 32594 |
This theorem is referenced by: threehalves 32620 hgt750lemd 34216 hgt750lem2 34220 |
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