deccl - Intuitionistic Logic Explorer

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Theorem deccl 9428

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 9415	. 2 ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
2		9nn0 9230	. . . 4 ⊢ 9 ∈ ℕ₀
3		1nn0 9222	. . . 4 ⊢ 1 ∈ ℕ₀
4	2, 3	nn0addcli 9243	. . 3 ⊢ (9 + 1) ∈ ℕ₀
5		deccl.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
6		deccl.2	. . 3 ⊢ 𝐵 ∈ ℕ₀
7	4, 5, 6	numcl 9426	. 2 ⊢ (((9 + 1) · 𝐴) + 𝐵) ∈ ℕ₀
8	1, 7	eqeltri 2262	1 ⊢ ;𝐴𝐵 ∈ ℕ₀

Colors of variables: wff set class

Syntax hints: ∈ wcel 2160 (class class class)co 5896 1c1 7842 + caddc 7844 · cmul 7846 9c9 9007 ℕ₀cn0 9206 cdc 9414

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-sub 8160 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-5 9011 df-6 9012 df-7 9013 df-8 9014 df-9 9015 df-n0 9207 df-dec 9415

This theorem is referenced by: 10nn0 9431 3declth 9445 3decltc 9446 decleh 9448 sq10 10724 3dvds2dec 11903 1kp2ke3k 14934