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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 9209 | . 2 ⊢ (𝑇 · 𝐴) ∈ ℕ0 |
4 | numnncl.3 | . 2 ⊢ 𝐵 ∈ ℕ | |
5 | nn0nnaddcl 9202 | . 2 ⊢ (((𝑇 · 𝐴) ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → ((𝑇 · 𝐴) + 𝐵) ∈ ℕ) | |
6 | 3, 4, 5 | mp2an 426 | 1 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 (class class class)co 5871 + caddc 7810 · cmul 7812 ℕcn 8914 ℕ0cn0 9171 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-setind 4535 ax-cnex 7898 ax-resscn 7899 ax-1cn 7900 ax-1re 7901 ax-icn 7902 ax-addcl 7903 ax-addrcl 7904 ax-mulcl 7905 ax-addcom 7907 ax-mulcom 7908 ax-addass 7909 ax-mulass 7910 ax-distr 7911 ax-i2m1 7912 ax-1rid 7914 ax-0id 7915 ax-rnegex 7916 ax-cnre 7918 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5176 df-fun 5216 df-fv 5222 df-riota 5827 df-ov 5874 df-oprab 5875 df-mpo 5876 df-sub 8125 df-inn 8915 df-n0 9172 |
This theorem is referenced by: decnncl 9398 |
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