deccl - Intuitionistic Logic Explorer

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Theorem deccl 8786

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 8773	. 2 ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵)
2		9nn0 8589	. . . 4 ⊢ 9 ∈ ℕ₀
3		1nn0 8581	. . . 4 ⊢ 1 ∈ ℕ₀
4	2, 3	nn0addcli 8602	. . 3 ⊢ (9 + 1) ∈ ℕ₀
5		deccl.1	. . 3 ⊢ 𝐴 ∈ ℕ₀
6		deccl.2	. . 3 ⊢ 𝐵 ∈ ℕ₀
7	4, 5, 6	numcl 8784	. 2 ⊢ (((9 + 1) · 𝐴) + 𝐵) ∈ ℕ₀
8	1, 7	eqeltri 2155	1 ⊢ ;𝐴𝐵 ∈ ℕ₀

Colors of variables: wff set class

Syntax hints: ∈ wcel 1434 (class class class)co 5591 1c1 7254 + caddc 7256 · cmul 7258 9c9 8373 ℕ₀cn0 8565 cdc 8772

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-setind 4316 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-addcom 7348 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-i2m1 7353 ax-1rid 7355 ax-0id 7356 ax-rnegex 7357 ax-cnre 7359

This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-iota 4934 df-fun 4971 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-sub 7558 df-inn 8317 df-2 8375 df-3 8376 df-4 8377 df-5 8378 df-6 8379 df-7 8380 df-8 8381 df-9 8382 df-n0 8566 df-dec 8773

This theorem is referenced by: 10nn0 8789 3declth 8803 3decltc 8804 decleh 8806 sq10 9956 3dvds2dec 10646 1kp2ke3k 10995