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Theorem zdcle 9556

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 9522	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
2		zre 9483	. . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
3		zre 9483	. . 3 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
4		ltle 8267	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵))
5		orc 719	. . . . . 6 ⊢ (𝐴 ≤ 𝐵 → (𝐴 ≤ 𝐵 ∨ ¬ 𝐴 ≤ 𝐵))
6		df-dc 842	. . . . . 6 ⊢ (DECID 𝐴 ≤ 𝐵 ↔ (𝐴 ≤ 𝐵 ∨ ¬ 𝐴 ≤ 𝐵))
7	5, 6	sylibr 134	. . . . 5 ⊢ (𝐴 ≤ 𝐵 → DECID 𝐴 ≤ 𝐵)
8	4, 7	syl6 33	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → DECID 𝐴 ≤ 𝐵))
9		eqle 8271	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵)
10	9, 7	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → DECID 𝐴 ≤ 𝐵)
11	10	ex 115	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 = 𝐵 → DECID 𝐴 ≤ 𝐵))
12	11	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → DECID 𝐴 ≤ 𝐵))
13		lenlt 8255	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
14	13	biimpd 144	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 → ¬ 𝐵 < 𝐴))
15	14	con2d 629	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 → ¬ 𝐴 ≤ 𝐵))
16		olc 718	. . . . . 6 ⊢ (¬ 𝐴 ≤ 𝐵 → (𝐴 ≤ 𝐵 ∨ ¬ 𝐴 ≤ 𝐵))
17	16, 6	sylibr 134	. . . . 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 1340	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → DECID 𝐴 ≤ 𝐵))
20	2, 3, 19	syl2an 289	. 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 104 ∨ wo 715 DECID wdc 841 ∨ w3o 1003 = wceq 1397 ∈ wcel 2202 class class class wbr 4088 ℝcr 8031 < clt 8214 ≤ cle 8215 ℤcz 9479

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-addcom 8132 ax-addass 8134 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-0id 8140 ax-rnegex 8141 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-ltadd 8148

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-br 4089 df-opab 4151 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-inn 9144 df-n0 9403 df-z 9480

This theorem is referenced by: uzin 9789 xnn0dcle 10037 nelfzo 10387 exfzdc 10487 infssuzex 10494 modfzo0difsn 10658 fzfig 10693 iseqf1olemjpcl 10771 iseqf1olemqpcl 10772 seq3f1oleml 10779 seq3f1o 10780 fser0const 10798 ccatsymb 11183 fzowrddc 11232 swrdnd 11244 swrdsbslen 11251 swrdspsleq 11252 pfxccat3 11319 swrdccat 11320 pfxccat3a 11323 swrdccat3blem 11324 swrdccat3b 11325 uzin2 11552 2zsupmax 11791 2zinfmin 11808 sumeq2 11924 summodclem2a 11947 fsum3 11953 fsumcl2lem 11964 fsumadd 11972 sumsnf 11975 fsummulc2 12014 explecnv 12071 prodeq2 12123 prodmodclem3 12141 prodmodclem2a 12142 fprodseq 12149 prod1dc 12152 fprodmul 12157 prodsnf 12158 pcdvdsb 12898 pcmpt2 12922 pcmptdvds 12923 pcprod 12924 pcfac 12928 1arithlem4 12944 plyaddlem1 15477 plyaddlem 15479 gsumgfsum 16711

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