zdclt - Intuitionistic Logic Explorer

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Theorem zdclt 9524

Description: Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)

**Assertion**
Ref	Expression
zdclt	⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 < 𝐵)

**Proof of Theorem zdclt**
Step	Hyp	Ref	Expression
1		ztri3or 9489	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
2		zre 9450	. . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
3		zre 9450	. . 3 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
4		orc 717	. . . . . 6 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
5		df-dc 840	. . . . . 6 ⊢ (DECID 𝐴 < 𝐵 ↔ (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
6	4, 5	sylibr 134	. . . . 5 ⊢ (𝐴 < 𝐵 → DECID 𝐴 < 𝐵)
7	6	a1i 9	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → DECID 𝐴 < 𝐵))
8		ltnr 8223	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
9	8	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
10		breq2 4087	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
11	10	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
12	9, 11	mtbid 676	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
13		olc 716	. . . . . . . 8 ⊢ (¬ 𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
14	13, 5	sylibr 134	. . . . . . 7 ⊢ (¬ 𝐴 < 𝐵 → DECID 𝐴 < 𝐵)
15	12, 14	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → DECID 𝐴 < 𝐵)
16	15	ex 115	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 = 𝐵 → DECID 𝐴 < 𝐵))
17	16	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → DECID 𝐴 < 𝐵))
18		ltnsym 8232	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
19	18	ancoms 268	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
20	19, 14	syl6 33	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 → DECID 𝐴 < 𝐵))
21	7, 17, 20	3jaod 1338	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → DECID 𝐴 < 𝐵))
22	2, 3, 21	syl2an 289	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → DECID 𝐴 < 𝐵))
23	1, 22	mpd 13	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 < 𝐵)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 713 DECID wdc 839 ∨ w3o 1001 = wceq 1395 ∈ wcel 2200 class class class wbr 4083 ℝcr 7998 < clt 8181 ℤcz 9446

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-addcom 8099 ax-addass 8101 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-0id 8107 ax-rnegex 8108 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-ltadd 8115

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-br 4084 df-opab 4146 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-inn 9111 df-n0 9370 df-z 9447

This theorem is referenced by: fztri3or 10235 modifeq2int 10608 modsumfzodifsn 10618 seqf1oglem1 10741 seqf1oglem2 10742 exp3val 10763 ccatsymb 11137 fzowrddc 11179 swrd0g 11192 cvgratz 12043 bitsfzolem 12465 bitsmod 12467 infpnlem1 12882 infpnlem2 12883 gsumfzval 13424 gsumfzz 13528 gsumfzcl 13532 mulgval 13659 mulgfng 13661 subgmulg 13725 gsumfzreidx 13874 gsumfzsubmcl 13875 gsumfzmptfidmadd 13876 gsumfzmhm 13880 gsumfzfsum 14552 lgsval 15683 lgscllem 15686 lgsneg 15703 lgsdilem 15706 lgsdir 15714 lgsdi 15716 lgsne0 15717 lgsquadlemsfi 15754 lgsquadlem3 15758

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