deccl - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > deccl		GIF version

Theorem deccl 9560

Description: Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)

**Hypotheses**
Ref	Expression
deccl.1	⊢ 𝐴 ∈ ℕ₀
deccl.2	⊢ 𝐵 ∈ ℕ₀

**Assertion**
Ref	Expression
deccl	⊢ ;𝐴𝐵 ∈ ℕ₀

**Proof of Theorem deccl**
Step	Hyp	Ref	Expression
1		df-dec 9547	. 2 ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
2		9nn0 9361	. . . 4 ⊢ 9 ∈ ℕ₀
3		1nn0 9353	. . . 4 ⊢ 1 ∈ ℕ₀
4	2, 3	nn0addcli 9374	. . 3 ⊢ (9 + 1) ∈ ℕ₀
5		deccl.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
6		deccl.2	. . 3 ⊢ 𝐵 ∈ ℕ₀
7	4, 5, 6	numcl 9558	. 2 ⊢ (((9 + 1) · 𝐴) + 𝐵) ∈ ℕ₀
8	1, 7	eqeltri 2282	1 ⊢ ;𝐴𝐵 ∈ ℕ₀

Colors of variables: wff set class

Syntax hints: ∈ wcel 2180 (class class class)co 5974 1c1 7968 + caddc 7970 · cmul 7972 9c9 9136 ℕ₀cn0 9337 cdc 9546

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 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-14 2183 ax-ext 2191 ax-sep 4181 ax-pow 4237 ax-pr 4272 ax-setind 4606 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-cnre 8078

This theorem depends on definitions: df-bi 117 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-ral 2493 df-rex 2494 df-reu 2495 df-rab 2497 df-v 2781 df-sbc 3009 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-br 4063 df-opab 4125 df-id 4361 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-iota 5254 df-fun 5296 df-fv 5302 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-sub 8287 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-5 9140 df-6 9141 df-7 9142 df-8 9143 df-9 9144 df-n0 9338 df-dec 9547

This theorem is referenced by: 10nn0 9563 3declth 9577 3decltc 9578 decleh 9580 sq10 10901 3dvds2dec 12343 dec2dvds 12900 dec5dvds2 12902 2exp8 12924 2exp11 12925 2exp16 12926 1kp2ke3k 15998