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Mirrors > Home > ILE Home > Th. List > decmulnc | GIF version |
Description: The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021.) |
Ref | Expression |
---|---|
decmulnc.n | ⊢ 𝑁 ∈ ℕ0 |
decmulnc.a | ⊢ 𝐴 ∈ ℕ0 |
decmulnc.b | ⊢ 𝐵 ∈ ℕ0 |
Ref | Expression |
---|---|
decmulnc | ⊢ (𝑁 · ;𝐴𝐵) = ;(𝑁 · 𝐴)(𝑁 · 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decmulnc.n | . 2 ⊢ 𝑁 ∈ ℕ0 | |
2 | decmulnc.a | . 2 ⊢ 𝐴 ∈ ℕ0 | |
3 | decmulnc.b | . 2 ⊢ 𝐵 ∈ ℕ0 | |
4 | eqid 2170 | . 2 ⊢ ;𝐴𝐵 = ;𝐴𝐵 | |
5 | 1, 3 | nn0mulcli 9162 | . 2 ⊢ (𝑁 · 𝐵) ∈ ℕ0 |
6 | 0nn0 9139 | . 2 ⊢ 0 ∈ ℕ0 | |
7 | 1, 2 | nn0mulcli 9162 | . . . 4 ⊢ (𝑁 · 𝐴) ∈ ℕ0 |
8 | 7 | nn0cni 9136 | . . 3 ⊢ (𝑁 · 𝐴) ∈ ℂ |
9 | 8 | addid1i 8050 | . 2 ⊢ ((𝑁 · 𝐴) + 0) = (𝑁 · 𝐴) |
10 | 5 | dec0h 9353 | . 2 ⊢ (𝑁 · 𝐵) = ;0(𝑁 · 𝐵) |
11 | 1, 2, 3, 4, 5, 6, 9, 10 | decmul2c 9397 | 1 ⊢ (𝑁 · ;𝐴𝐵) = ;(𝑁 · 𝐴)(𝑁 · 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 (class class class)co 5851 0cc0 7763 · cmul 7768 ℕ0cn0 9124 ;cdc 9332 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-cnre 7874 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-sub 8081 df-inn 8868 df-2 8926 df-3 8927 df-4 8928 df-5 8929 df-6 8930 df-7 8931 df-8 8932 df-9 8933 df-n0 9125 df-dec 9333 |
This theorem is referenced by: 11multnc 9399 |
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