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Mirrors > Home > ILE Home > Th. List > nummul1c | GIF version |
Description: The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
nummul1c.1 | ⊢ 𝑇 ∈ ℕ0 |
nummul1c.2 | ⊢ 𝑃 ∈ ℕ0 |
nummul1c.3 | ⊢ A ∈ ℕ0 |
nummul1c.4 | ⊢ B ∈ ℕ0 |
nummul1c.5 | ⊢ 𝑁 = ((𝑇 · A) + B) |
nummul1c.6 | ⊢ 𝐷 ∈ ℕ0 |
nummul1c.7 | ⊢ 𝐸 ∈ ℕ0 |
nummul1c.8 | ⊢ ((A · 𝑃) + 𝐸) = 𝐶 |
nummul1c.9 | ⊢ (B · 𝑃) = ((𝑇 · 𝐸) + 𝐷) |
Ref | Expression |
---|---|
nummul1c | ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nummul1c.5 | . . . 4 ⊢ 𝑁 = ((𝑇 · A) + B) | |
2 | nummul1c.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 | |
3 | nummul1c.3 | . . . . 5 ⊢ A ∈ ℕ0 | |
4 | nummul1c.4 | . . . . 5 ⊢ B ∈ ℕ0 | |
5 | 2, 3, 4 | numcl 8154 | . . . 4 ⊢ ((𝑇 · A) + B) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2107 | . . 3 ⊢ 𝑁 ∈ ℕ0 |
7 | nummul1c.2 | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
8 | 6, 7 | num0u 8152 | . 2 ⊢ (𝑁 · 𝑃) = ((𝑁 · 𝑃) + 0) |
9 | 0nn0 7972 | . . 3 ⊢ 0 ∈ ℕ0 | |
10 | 2, 9 | num0h 8153 | . . 3 ⊢ 0 = ((𝑇 · 0) + 0) |
11 | nummul1c.6 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
12 | nummul1c.7 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
13 | 12 | nn0cni 7969 | . . . . . 6 ⊢ 𝐸 ∈ ℂ |
14 | 13 | addid2i 6953 | . . . . 5 ⊢ (0 + 𝐸) = 𝐸 |
15 | 14 | oveq2i 5466 | . . . 4 ⊢ ((A · 𝑃) + (0 + 𝐸)) = ((A · 𝑃) + 𝐸) |
16 | nummul1c.8 | . . . 4 ⊢ ((A · 𝑃) + 𝐸) = 𝐶 | |
17 | 15, 16 | eqtri 2057 | . . 3 ⊢ ((A · 𝑃) + (0 + 𝐸)) = 𝐶 |
18 | 4, 7 | num0u 8152 | . . . 4 ⊢ (B · 𝑃) = ((B · 𝑃) + 0) |
19 | nummul1c.9 | . . . 4 ⊢ (B · 𝑃) = ((𝑇 · 𝐸) + 𝐷) | |
20 | 18, 19 | eqtr3i 2059 | . . 3 ⊢ ((B · 𝑃) + 0) = ((𝑇 · 𝐸) + 𝐷) |
21 | 2, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20 | nummac 8175 | . 2 ⊢ ((𝑁 · 𝑃) + 0) = ((𝑇 · 𝐶) + 𝐷) |
22 | 8, 21 | eqtri 2057 | 1 ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∈ wcel 1390 (class class class)co 5455 0cc0 6711 + caddc 6714 · cmul 6716 ℕ0cn0 7957 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-mulcom 6784 ax-addass 6785 ax-mulass 6786 ax-distr 6787 ax-i2m1 6788 ax-1rid 6790 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-sub 6981 df-inn 7696 df-n0 7958 |
This theorem is referenced by: nummul2c 8180 decmul1c 8190 |
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