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Mirrors > Home > ILE Home > Th. List > decmul1 | GIF version |
Description: The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decmul1.p | ⊢ 𝑃 ∈ ℕ0 |
decmul1.a | ⊢ 𝐴 ∈ ℕ0 |
decmul1.b | ⊢ 𝐵 ∈ ℕ0 |
decmul1.n | ⊢ 𝑁 = ;𝐴𝐵 |
decmul1.0 | ⊢ 𝐷 ∈ ℕ0 |
decmul1.c | ⊢ (𝐴 · 𝑃) = 𝐶 |
decmul1.d | ⊢ (𝐵 · 𝑃) = 𝐷 |
Ref | Expression |
---|---|
decmul1 | ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 9199 | . . 3 ⊢ ;10 ∈ ℕ0 | |
2 | decmul1.p | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
3 | decmul1.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
4 | decmul1.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
5 | decmul1.n | . . . 4 ⊢ 𝑁 = ;𝐴𝐵 | |
6 | dfdec10 9185 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
7 | 5, 6 | eqtri 2160 | . . 3 ⊢ 𝑁 = ((;10 · 𝐴) + 𝐵) |
8 | decmul1.0 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
9 | 0nn0 8992 | . . 3 ⊢ 0 ∈ ℕ0 | |
10 | 3, 2 | nn0mulcli 9015 | . . . . . 6 ⊢ (𝐴 · 𝑃) ∈ ℕ0 |
11 | 10 | nn0cni 8989 | . . . . 5 ⊢ (𝐴 · 𝑃) ∈ ℂ |
12 | 11 | addid1i 7904 | . . . 4 ⊢ ((𝐴 · 𝑃) + 0) = (𝐴 · 𝑃) |
13 | decmul1.c | . . . 4 ⊢ (𝐴 · 𝑃) = 𝐶 | |
14 | 12, 13 | eqtri 2160 | . . 3 ⊢ ((𝐴 · 𝑃) + 0) = 𝐶 |
15 | decmul1.d | . . . . 5 ⊢ (𝐵 · 𝑃) = 𝐷 | |
16 | 15 | oveq2i 5785 | . . . 4 ⊢ (0 + (𝐵 · 𝑃)) = (0 + 𝐷) |
17 | 4, 2 | nn0mulcli 9015 | . . . . . 6 ⊢ (𝐵 · 𝑃) ∈ ℕ0 |
18 | 17 | nn0cni 8989 | . . . . 5 ⊢ (𝐵 · 𝑃) ∈ ℂ |
19 | 18 | addid2i 7905 | . . . 4 ⊢ (0 + (𝐵 · 𝑃)) = (𝐵 · 𝑃) |
20 | 1 | nn0cni 8989 | . . . . . . 7 ⊢ ;10 ∈ ℂ |
21 | 20 | mul01i 8153 | . . . . . 6 ⊢ (;10 · 0) = 0 |
22 | 21 | eqcomi 2143 | . . . . 5 ⊢ 0 = (;10 · 0) |
23 | 22 | oveq1i 5784 | . . . 4 ⊢ (0 + 𝐷) = ((;10 · 0) + 𝐷) |
24 | 16, 19, 23 | 3eqtr3i 2168 | . . 3 ⊢ (𝐵 · 𝑃) = ((;10 · 0) + 𝐷) |
25 | 1, 2, 3, 4, 7, 8, 9, 14, 24 | nummul1c 9230 | . 2 ⊢ (𝑁 · 𝑃) = ((;10 · 𝐶) + 𝐷) |
26 | dfdec10 9185 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
27 | 25, 26 | eqtr4i 2163 | 1 ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 (class class class)co 5774 0cc0 7620 1c1 7621 + caddc 7623 · cmul 7625 ℕ0cn0 8977 ;cdc 9182 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-5 8782 df-6 8783 df-7 8784 df-8 8785 df-9 8786 df-n0 8978 df-dec 9183 |
This theorem is referenced by: sq10 10459 |
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