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Mirrors > Home > ILE Home > Th. List > decmul10add | GIF version |
Description: A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decmul10add.1 | ⊢ 𝐴 ∈ ℕ0 |
decmul10add.2 | ⊢ 𝐵 ∈ ℕ0 |
decmul10add.3 | ⊢ 𝑀 ∈ ℕ0 |
decmul10add.4 | ⊢ 𝐸 = (𝑀 · 𝐴) |
decmul10add.5 | ⊢ 𝐹 = (𝑀 · 𝐵) |
Ref | Expression |
---|---|
decmul10add | ⊢ (𝑀 · ;𝐴𝐵) = (;𝐸0 + 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdec10 8480 | . . 3 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
2 | 1 | oveq2i 5543 | . 2 ⊢ (𝑀 · ;𝐴𝐵) = (𝑀 · ((;10 · 𝐴) + 𝐵)) |
3 | decmul10add.3 | . . . 4 ⊢ 𝑀 ∈ ℕ0 | |
4 | 3 | nn0cni 8300 | . . 3 ⊢ 𝑀 ∈ ℂ |
5 | 10nn0 8494 | . . . . 5 ⊢ ;10 ∈ ℕ0 | |
6 | decmul10add.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
7 | 5, 6 | nn0mulcli 8326 | . . . 4 ⊢ (;10 · 𝐴) ∈ ℕ0 |
8 | 7 | nn0cni 8300 | . . 3 ⊢ (;10 · 𝐴) ∈ ℂ |
9 | decmul10add.2 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
10 | 9 | nn0cni 8300 | . . 3 ⊢ 𝐵 ∈ ℂ |
11 | 4, 8, 10 | adddii 7129 | . 2 ⊢ (𝑀 · ((;10 · 𝐴) + 𝐵)) = ((𝑀 · (;10 · 𝐴)) + (𝑀 · 𝐵)) |
12 | 5 | nn0cni 8300 | . . . . 5 ⊢ ;10 ∈ ℂ |
13 | 6 | nn0cni 8300 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
14 | 4, 12, 13 | mul12i 7254 | . . . 4 ⊢ (𝑀 · (;10 · 𝐴)) = (;10 · (𝑀 · 𝐴)) |
15 | 3, 6 | nn0mulcli 8326 | . . . . 5 ⊢ (𝑀 · 𝐴) ∈ ℕ0 |
16 | 15 | dec0u 8497 | . . . 4 ⊢ (;10 · (𝑀 · 𝐴)) = ;(𝑀 · 𝐴)0 |
17 | decmul10add.4 | . . . . . 6 ⊢ 𝐸 = (𝑀 · 𝐴) | |
18 | 17 | eqcomi 2085 | . . . . 5 ⊢ (𝑀 · 𝐴) = 𝐸 |
19 | 18 | deceq1i 8483 | . . . 4 ⊢ ;(𝑀 · 𝐴)0 = ;𝐸0 |
20 | 14, 16, 19 | 3eqtri 2105 | . . 3 ⊢ (𝑀 · (;10 · 𝐴)) = ;𝐸0 |
21 | decmul10add.5 | . . . 4 ⊢ 𝐹 = (𝑀 · 𝐵) | |
22 | 21 | eqcomi 2085 | . . 3 ⊢ (𝑀 · 𝐵) = 𝐹 |
23 | 20, 22 | oveq12i 5544 | . 2 ⊢ ((𝑀 · (;10 · 𝐴)) + (𝑀 · 𝐵)) = (;𝐸0 + 𝐹) |
24 | 2, 11, 23 | 3eqtri 2105 | 1 ⊢ (𝑀 · ;𝐴𝐵) = (;𝐸0 + 𝐹) |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∈ wcel 1433 (class class class)co 5532 0cc0 6981 1c1 6982 + caddc 6984 · cmul 6986 ℕ0cn0 8288 ;cdc 8477 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-5 8101 df-6 8102 df-7 8103 df-8 8104 df-9 8105 df-n0 8289 df-dec 8478 |
This theorem is referenced by: (None) |
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