deccl - Intuitionistic Logic Explorer

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Theorem deccl 8990

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 8977	. 2 ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
2		9nn0 8795	. . . 4 ⊢ 9 ∈ ℕ₀
3		1nn0 8787	. . . 4 ⊢ 1 ∈ ℕ₀
4	2, 3	nn0addcli 8808	. . 3 ⊢ (9 + 1) ∈ ℕ₀
5		deccl.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
6		deccl.2	. . 3 ⊢ 𝐵 ∈ ℕ₀
7	4, 5, 6	numcl 8988	. 2 ⊢ (((9 + 1) · 𝐴) + 𝐵) ∈ ℕ₀
8	1, 7	eqeltri 2167	1 ⊢ ;𝐴𝐵 ∈ ℕ₀

Colors of variables: wff set class

Syntax hints: ∈ wcel 1445 (class class class)co 5690 1c1 7448 + caddc 7450 · cmul 7452 9c9 8578 ℕ₀cn0 8771 cdc 8976

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553

This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-sub 7752 df-inn 8521 df-2 8579 df-3 8580 df-4 8581 df-5 8582 df-6 8583 df-7 8584 df-8 8585 df-9 8586 df-n0 8772 df-dec 8977

This theorem is referenced by: 10nn0 8993 3declth 9007 3decltc 9008 decleh 9010 sq10 10236 3dvds2dec 11293 1kp2ke3k 12359