zdcle - Intuitionistic Logic Explorer

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Theorem zdcle 8921

Description: Integer ≤ is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)

**Assertion**
Ref	Expression
zdcle	⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 ≤ 𝐵)

**Proof of Theorem zdcle**
Step	Hyp	Ref	Expression
1		ztri3or 8891	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
2		zre 8852	. . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
3		zre 8852	. . 3 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
4		ltle 7669	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
5		orc 671	. . . . . 6 ⊢ (𝐴 ≤ 𝐵 → (𝐴 ≤ 𝐵 ∨ ¬ 𝐴 ≤ 𝐵))
6		df-dc 784	. . . . . 6 ⊢ (DECID 𝐴 ≤ 𝐵 ↔ (𝐴 ≤ 𝐵 ∨ ¬ 𝐴 ≤ 𝐵))
7	5, 6	sylibr 133	. . . . 5 ⊢ (𝐴 ≤ 𝐵 → DECID 𝐴 ≤ 𝐵)
8	4, 7	syl6 33	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → DECID 𝐴 ≤ 𝐵))
9		eqle 7673	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵)
10	9, 7	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → DECID 𝐴 ≤ 𝐵)
11	10	ex 114	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 = 𝐵 → DECID 𝐴 ≤ 𝐵))
12	11	adantr 271	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → DECID 𝐴 ≤ 𝐵))
13		lenlt 7658	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
14	13	biimpd 143	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 → ¬ 𝐵 < 𝐴))
15	14	con2d 592	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 → ¬ 𝐴 ≤ 𝐵))
16		olc 670	. . . . . 6 ⊢ (¬ 𝐴 ≤ 𝐵 → (𝐴 ≤ 𝐵 ∨ ¬ 𝐴 ≤ 𝐵))
17	16, 6	sylibr 133	. . . . 5 ⊢ (¬ 𝐴 ≤ 𝐵 → DECID 𝐴 ≤ 𝐵)
18	15, 17	syl6 33	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 → DECID 𝐴 ≤ 𝐵))
19	8, 12, 18	3jaod 1247	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → DECID 𝐴 ≤ 𝐵))
20	2, 3, 19	syl2an 284	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → DECID 𝐴 ≤ 𝐵))
21	1, 20	mpd 13	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 ≤ 𝐵)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 667 DECID wdc 783 ∨ w3o 926 = wceq 1296 ∈ wcel 1445 class class class wbr 3867 ℝcr 7446 < clt 7619 ≤ cle 7620 ℤcz 8848

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-13 1456 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-un 4284 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-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-ltadd 7558

This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 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-nel 2358 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-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-inn 8521 df-n0 8772 df-z 8849

This theorem is referenced by: uzin 9150 exfzdc 9800 modfzo0difsn 9951 fzfig 9986 iseqf1olemjpcl 10061 iseqf1olemqpcl 10062 seq3f1oleml 10069 seq3f1o 10070 fser0const 10082 uzin2 10551 2zsupmax 10788 sumeq2 10917 summodclem2a 10939 fsum3 10945 fsumcl2lem 10956 fsumadd 10964 sumsnf 10967 fsummulc2 11006 explecnv 11063 infssuzex 11387

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