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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 8378 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2110 | . . 3 ⊢ 𝑁 ∈ ℕ0 |
7 | nummul1c.2 | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
8 | 6, 7 | num0u 8376 | . 2 ⊢ (𝑁 · 𝑃) = ((𝑁 · 𝑃) + 0) |
9 | 0nn0 8196 | . . 3 ⊢ 0 ∈ ℕ0 | |
10 | 2, 9 | num0h 8377 | . . 3 ⊢ 0 = ((𝑇 · 0) + 0) |
11 | nummul1c.6 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
12 | nummul1c.7 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
13 | 12 | nn0cni 8193 | . . . . . 6 ⊢ 𝐸 ∈ ℂ |
14 | 13 | addid2i 7156 | . . . . 5 ⊢ (0 + 𝐸) = 𝐸 |
15 | 14 | oveq2i 5523 | . . . 4 ⊢ ((𝐴 · 𝑃) + (0 + 𝐸)) = ((𝐴 · 𝑃) + 𝐸) |
16 | nummul1c.8 | . . . 4 ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 | |
17 | 15, 16 | eqtri 2060 | . . 3 ⊢ ((𝐴 · 𝑃) + (0 + 𝐸)) = 𝐶 |
18 | 4, 7 | num0u 8376 | . . . 4 ⊢ (𝐵 · 𝑃) = ((𝐵 · 𝑃) + 0) |
19 | nummul1c.9 | . . . 4 ⊢ (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷) | |
20 | 18, 19 | eqtr3i 2062 | . . 3 ⊢ ((𝐵 · 𝑃) + 0) = ((𝑇 · 𝐸) + 𝐷) |
21 | 2, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20 | nummac 8399 | . 2 ⊢ ((𝑁 · 𝑃) + 0) = ((𝑇 · 𝐶) + 𝐷) |
22 | 8, 21 | eqtri 2060 | 1 ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 (class class class)co 5512 0cc0 6889 + caddc 6892 · cmul 6894 ℕ0cn0 8181 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-inn 7915 df-n0 8182 |
This theorem is referenced by: nummul2c 8404 decmul1c 8414 |
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