deccl - Intuitionistic Logic Explorer

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Theorem deccl 9774

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 9761	. 2 ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
2		9nn0 9570	. . . 4 ⊢ 9 ∈ ℕ₀
3		1nn0 9562	. . . 4 ⊢ 1 ∈ ℕ₀
4	2, 3	nn0addcli 9583	. . 3 ⊢ (9 + 1) ∈ ℕ₀
5		deccl.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
6		deccl.2	. . 3 ⊢ 𝐵 ∈ ℕ₀
7	4, 5, 6	numcl 9772	. 2 ⊢ (((9 + 1) · 𝐴) + 𝐵) ∈ ℕ₀
8	1, 7	eqeltri 2311	1 ⊢ ;𝐴𝐵 ∈ ℕ₀

Colors of variables: wff set class

Syntax hints: ∈ wcel 2209 (class class class)co 6079 1c1 8174 + caddc 8176 · cmul 8178 9c9 9345 ℕ₀cn0 9546 cdc 9760

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4247 ax-pow 4309 ax-pr 4344 ax-setind 4682 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-cnre 8284

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-br 4129 df-opab 4191 df-id 4436 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-iota 5335 df-fun 5377 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-sub 8493 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-5 9349 df-6 9350 df-7 9351 df-8 9352 df-9 9353 df-n0 9547 df-dec 9761

This theorem is referenced by: 10nn0 9777 3declth 9791 3decltc 9792 decleh 9794 sq10 11133 3dvds2dec 12616 dec2dvds 13173 dec5dvds2 13175 2exp8 13197 2exp11 13198 2exp16 13199 log2ublem3 16068 log2ublog2 16069 birthdaylog2 16073 1kp2ke3k 16721