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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 9422 | . . 3 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
2 | 1 | oveq2i 5911 | . 2 ⊢ (𝑀 · ;𝐴𝐵) = (𝑀 · ((;10 · 𝐴) + 𝐵)) |
3 | decmul10add.3 | . . . 4 ⊢ 𝑀 ∈ ℕ0 | |
4 | 3 | nn0cni 9223 | . . 3 ⊢ 𝑀 ∈ ℂ |
5 | 10nn0 9436 | . . . . 5 ⊢ ;10 ∈ ℕ0 | |
6 | decmul10add.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
7 | 5, 6 | nn0mulcli 9249 | . . . 4 ⊢ (;10 · 𝐴) ∈ ℕ0 |
8 | 7 | nn0cni 9223 | . . 3 ⊢ (;10 · 𝐴) ∈ ℂ |
9 | decmul10add.2 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
10 | 9 | nn0cni 9223 | . . 3 ⊢ 𝐵 ∈ ℂ |
11 | 4, 8, 10 | adddii 8002 | . 2 ⊢ (𝑀 · ((;10 · 𝐴) + 𝐵)) = ((𝑀 · (;10 · 𝐴)) + (𝑀 · 𝐵)) |
12 | 5 | nn0cni 9223 | . . . . 5 ⊢ ;10 ∈ ℂ |
13 | 6 | nn0cni 9223 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
14 | 4, 12, 13 | mul12i 8138 | . . . 4 ⊢ (𝑀 · (;10 · 𝐴)) = (;10 · (𝑀 · 𝐴)) |
15 | 3, 6 | nn0mulcli 9249 | . . . . 5 ⊢ (𝑀 · 𝐴) ∈ ℕ0 |
16 | 15 | dec0u 9439 | . . . 4 ⊢ (;10 · (𝑀 · 𝐴)) = ;(𝑀 · 𝐴)0 |
17 | decmul10add.4 | . . . . . 6 ⊢ 𝐸 = (𝑀 · 𝐴) | |
18 | 17 | eqcomi 2193 | . . . . 5 ⊢ (𝑀 · 𝐴) = 𝐸 |
19 | 18 | deceq1i 9425 | . . . 4 ⊢ ;(𝑀 · 𝐴)0 = ;𝐸0 |
20 | 14, 16, 19 | 3eqtri 2214 | . . 3 ⊢ (𝑀 · (;10 · 𝐴)) = ;𝐸0 |
21 | decmul10add.5 | . . . 4 ⊢ 𝐹 = (𝑀 · 𝐵) | |
22 | 21 | eqcomi 2193 | . . 3 ⊢ (𝑀 · 𝐵) = 𝐹 |
23 | 20, 22 | oveq12i 5912 | . 2 ⊢ ((𝑀 · (;10 · 𝐴)) + (𝑀 · 𝐵)) = (;𝐸0 + 𝐹) |
24 | 2, 11, 23 | 3eqtri 2214 | 1 ⊢ (𝑀 · ;𝐴𝐵) = (;𝐸0 + 𝐹) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 (class class class)co 5900 0cc0 7846 1c1 7847 + caddc 7849 · cmul 7851 ℕ0cn0 9211 ;cdc 9419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-cnre 7957 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-iota 5199 df-fun 5240 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-sub 8165 df-inn 8955 df-2 9013 df-3 9014 df-4 9015 df-5 9016 df-6 9017 df-7 9018 df-8 9019 df-9 9020 df-n0 9212 df-dec 9420 |
This theorem is referenced by: (None) |
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