Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > numnncl | GIF version |
Description: Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
numnncl.1 | ⊢ 𝑇 ∈ ℕ0 |
numnncl.2 | ⊢ 𝐴 ∈ ℕ0 |
numnncl.3 | ⊢ 𝐵 ∈ ℕ |
Ref | Expression |
---|---|
numnncl | ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numnncl.1 | . . 3 ⊢ 𝑇 ∈ ℕ0 | |
2 | numnncl.2 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
3 | 1, 2 | nn0mulcli 8326 | . 2 ⊢ (𝑇 · 𝐴) ∈ ℕ0 |
4 | numnncl.3 | . 2 ⊢ 𝐵 ∈ ℕ | |
5 | nn0nnaddcl 8319 | . 2 ⊢ (((𝑇 · 𝐴) ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → ((𝑇 · 𝐴) + 𝐵) ∈ ℕ) | |
6 | 3, 4, 5 | mp2an 416 | 1 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1433 (class class class)co 5532 + caddc 6984 · cmul 6986 ℕcn 8039 ℕ0cn0 8288 |
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-n0 8289 |
This theorem is referenced by: decnncl 8496 |
Copyright terms: Public domain | W3C validator |