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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 9209 | . . 3 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
2 | 1 | oveq2i 5793 | . 2 ⊢ (𝑀 · ;𝐴𝐵) = (𝑀 · ((;10 · 𝐴) + 𝐵)) |
3 | decmul10add.3 | . . . 4 ⊢ 𝑀 ∈ ℕ0 | |
4 | 3 | nn0cni 9013 | . . 3 ⊢ 𝑀 ∈ ℂ |
5 | 10nn0 9223 | . . . . 5 ⊢ ;10 ∈ ℕ0 | |
6 | decmul10add.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
7 | 5, 6 | nn0mulcli 9039 | . . . 4 ⊢ (;10 · 𝐴) ∈ ℕ0 |
8 | 7 | nn0cni 9013 | . . 3 ⊢ (;10 · 𝐴) ∈ ℂ |
9 | decmul10add.2 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
10 | 9 | nn0cni 9013 | . . 3 ⊢ 𝐵 ∈ ℂ |
11 | 4, 8, 10 | adddii 7800 | . 2 ⊢ (𝑀 · ((;10 · 𝐴) + 𝐵)) = ((𝑀 · (;10 · 𝐴)) + (𝑀 · 𝐵)) |
12 | 5 | nn0cni 9013 | . . . . 5 ⊢ ;10 ∈ ℂ |
13 | 6 | nn0cni 9013 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
14 | 4, 12, 13 | mul12i 7932 | . . . 4 ⊢ (𝑀 · (;10 · 𝐴)) = (;10 · (𝑀 · 𝐴)) |
15 | 3, 6 | nn0mulcli 9039 | . . . . 5 ⊢ (𝑀 · 𝐴) ∈ ℕ0 |
16 | 15 | dec0u 9226 | . . . 4 ⊢ (;10 · (𝑀 · 𝐴)) = ;(𝑀 · 𝐴)0 |
17 | decmul10add.4 | . . . . . 6 ⊢ 𝐸 = (𝑀 · 𝐴) | |
18 | 17 | eqcomi 2144 | . . . . 5 ⊢ (𝑀 · 𝐴) = 𝐸 |
19 | 18 | deceq1i 9212 | . . . 4 ⊢ ;(𝑀 · 𝐴)0 = ;𝐸0 |
20 | 14, 16, 19 | 3eqtri 2165 | . . 3 ⊢ (𝑀 · (;10 · 𝐴)) = ;𝐸0 |
21 | decmul10add.5 | . . . 4 ⊢ 𝐹 = (𝑀 · 𝐵) | |
22 | 21 | eqcomi 2144 | . . 3 ⊢ (𝑀 · 𝐵) = 𝐹 |
23 | 20, 22 | oveq12i 5794 | . 2 ⊢ ((𝑀 · (;10 · 𝐴)) + (𝑀 · 𝐵)) = (;𝐸0 + 𝐹) |
24 | 2, 11, 23 | 3eqtri 2165 | 1 ⊢ (𝑀 · ;𝐴𝐵) = (;𝐸0 + 𝐹) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 (class class class)co 5782 0cc0 7644 1c1 7645 + caddc 7647 · cmul 7649 ℕ0cn0 9001 ;cdc 9206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-n0 9002 df-dec 9207 |
This theorem is referenced by: (None) |
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