![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > deccl | GIF version |
Description: Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
deccl.1 | ⊢ 𝐴 ∈ ℕ0 |
deccl.2 | ⊢ 𝐵 ∈ ℕ0 |
Ref | Expression |
---|---|
deccl | ⊢ ;𝐴𝐵 ∈ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dec 9207 | . 2 ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) | |
2 | 9nn0 9025 | . . . 4 ⊢ 9 ∈ ℕ0 | |
3 | 1nn0 9017 | . . . 4 ⊢ 1 ∈ ℕ0 | |
4 | 2, 3 | nn0addcli 9038 | . . 3 ⊢ (9 + 1) ∈ ℕ0 |
5 | deccl.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
6 | deccl.2 | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
7 | 4, 5, 6 | numcl 9218 | . 2 ⊢ (((9 + 1) · 𝐴) + 𝐵) ∈ ℕ0 |
8 | 1, 7 | eqeltri 2213 | 1 ⊢ ;𝐴𝐵 ∈ ℕ0 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 (class class class)co 5782 1c1 7645 + caddc 7647 · cmul 7649 9c9 8802 ℕ0cn0 9001 ;cdc 9206 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-n0 9002 df-dec 9207 |
This theorem is referenced by: 10nn0 9223 3declth 9237 3decltc 9238 decleh 9240 sq10 10490 3dvds2dec 11599 1kp2ke3k 13107 |
Copyright terms: Public domain | W3C validator |