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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 9330 | . . 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 9316 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
7 | 5, 6 | eqtri 2185 | . . 3 ⊢ 𝑁 = ((;10 · 𝐴) + 𝐵) |
8 | decmul1.0 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
9 | 0nn0 9120 | . . 3 ⊢ 0 ∈ ℕ0 | |
10 | 3, 2 | nn0mulcli 9143 | . . . . . 6 ⊢ (𝐴 · 𝑃) ∈ ℕ0 |
11 | 10 | nn0cni 9117 | . . . . 5 ⊢ (𝐴 · 𝑃) ∈ ℂ |
12 | 11 | addid1i 8031 | . . . 4 ⊢ ((𝐴 · 𝑃) + 0) = (𝐴 · 𝑃) |
13 | decmul1.c | . . . 4 ⊢ (𝐴 · 𝑃) = 𝐶 | |
14 | 12, 13 | eqtri 2185 | . . 3 ⊢ ((𝐴 · 𝑃) + 0) = 𝐶 |
15 | decmul1.d | . . . . 5 ⊢ (𝐵 · 𝑃) = 𝐷 | |
16 | 15 | oveq2i 5847 | . . . 4 ⊢ (0 + (𝐵 · 𝑃)) = (0 + 𝐷) |
17 | 4, 2 | nn0mulcli 9143 | . . . . . 6 ⊢ (𝐵 · 𝑃) ∈ ℕ0 |
18 | 17 | nn0cni 9117 | . . . . 5 ⊢ (𝐵 · 𝑃) ∈ ℂ |
19 | 18 | addid2i 8032 | . . . 4 ⊢ (0 + (𝐵 · 𝑃)) = (𝐵 · 𝑃) |
20 | 1 | nn0cni 9117 | . . . . . . 7 ⊢ ;10 ∈ ℂ |
21 | 20 | mul01i 8280 | . . . . . 6 ⊢ (;10 · 0) = 0 |
22 | 21 | eqcomi 2168 | . . . . 5 ⊢ 0 = (;10 · 0) |
23 | 22 | oveq1i 5846 | . . . 4 ⊢ (0 + 𝐷) = ((;10 · 0) + 𝐷) |
24 | 16, 19, 23 | 3eqtr3i 2193 | . . 3 ⊢ (𝐵 · 𝑃) = ((;10 · 0) + 𝐷) |
25 | 1, 2, 3, 4, 7, 8, 9, 14, 24 | nummul1c 9361 | . 2 ⊢ (𝑁 · 𝑃) = ((;10 · 𝐶) + 𝐷) |
26 | dfdec10 9316 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
27 | 25, 26 | eqtr4i 2188 | 1 ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 (class class class)co 5836 0cc0 7744 1c1 7745 + caddc 7747 · cmul 7749 ℕ0cn0 9105 ;cdc 9313 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-n0 9106 df-dec 9314 |
This theorem is referenced by: sq10 10614 |
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