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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 8494 | . . 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 8480 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
7 | 5, 6 | eqtri 2101 | . . 3 ⊢ 𝑁 = ((;10 · 𝐴) + 𝐵) |
8 | decmul1.0 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
9 | 0nn0 8303 | . . 3 ⊢ 0 ∈ ℕ0 | |
10 | 3, 2 | nn0mulcli 8326 | . . . . . 6 ⊢ (𝐴 · 𝑃) ∈ ℕ0 |
11 | 10 | nn0cni 8300 | . . . . 5 ⊢ (𝐴 · 𝑃) ∈ ℂ |
12 | 11 | addid1i 7250 | . . . 4 ⊢ ((𝐴 · 𝑃) + 0) = (𝐴 · 𝑃) |
13 | decmul1.c | . . . 4 ⊢ (𝐴 · 𝑃) = 𝐶 | |
14 | 12, 13 | eqtri 2101 | . . 3 ⊢ ((𝐴 · 𝑃) + 0) = 𝐶 |
15 | decmul1.d | . . . . 5 ⊢ (𝐵 · 𝑃) = 𝐷 | |
16 | 15 | oveq2i 5543 | . . . 4 ⊢ (0 + (𝐵 · 𝑃)) = (0 + 𝐷) |
17 | 4, 2 | nn0mulcli 8326 | . . . . . 6 ⊢ (𝐵 · 𝑃) ∈ ℕ0 |
18 | 17 | nn0cni 8300 | . . . . 5 ⊢ (𝐵 · 𝑃) ∈ ℂ |
19 | 18 | addid2i 7251 | . . . 4 ⊢ (0 + (𝐵 · 𝑃)) = (𝐵 · 𝑃) |
20 | 1 | nn0cni 8300 | . . . . . . 7 ⊢ ;10 ∈ ℂ |
21 | 20 | mul01i 7495 | . . . . . 6 ⊢ (;10 · 0) = 0 |
22 | 21 | eqcomi 2085 | . . . . 5 ⊢ 0 = (;10 · 0) |
23 | 22 | oveq1i 5542 | . . . 4 ⊢ (0 + 𝐷) = ((;10 · 0) + 𝐷) |
24 | 16, 19, 23 | 3eqtr3i 2109 | . . 3 ⊢ (𝐵 · 𝑃) = ((;10 · 0) + 𝐷) |
25 | 1, 2, 3, 4, 7, 8, 9, 14, 24 | nummul1c 8525 | . 2 ⊢ (𝑁 · 𝑃) = ((;10 · 𝐶) + 𝐷) |
26 | dfdec10 8480 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
27 | 25, 26 | eqtr4i 2104 | 1 ⊢ (𝑁 · 𝑃) = ;𝐶𝐷 |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∈ wcel 1433 (class class class)co 5532 0cc0 6981 1c1 6982 + caddc 6984 · cmul 6986 ℕ0cn0 8288 ;cdc 8477 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-5 8101 df-6 8102 df-7 8103 df-8 8104 df-9 8105 df-n0 8289 df-dec 8478 |
This theorem is referenced by: sq10 9640 |
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