zdclt - Intuitionistic Logic Explorer

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Theorem zdclt 9450

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 9415	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
2		zre 9376	. . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
3		zre 9376	. . 3 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
4		orc 714	. . . . . 6 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵))
5		df-dc 837	. . . . . 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 8149	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
9	8	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
10		breq2 4048	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
11	10	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
12	9, 11	mtbid 674	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
13		olc 713	. . . . . . . 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 8158	. . . . . 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 1317	. . 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 710 DECID wdc 836 ∨ w3o 980 = wceq 1373 ∈ wcel 2176 class class class wbr 4044 ℝcr 7924 < clt 8107 ℤcz 9372

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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-addcom 8025 ax-addass 8027 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-0id 8033 ax-rnegex 8034 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-ltadd 8041

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-br 4045 df-opab 4106 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-iota 5232 df-fun 5273 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-inn 9037 df-n0 9296 df-z 9373

This theorem is referenced by: fztri3or 10161 modifeq2int 10531 modsumfzodifsn 10541 seqf1oglem1 10664 seqf1oglem2 10665 exp3val 10686 ccatsymb 11058 fzowrddc 11100 swrd0g 11113 cvgratz 11843 bitsfzolem 12265 bitsmod 12267 infpnlem1 12682 infpnlem2 12683 gsumfzval 13223 gsumfzz 13327 gsumfzcl 13331 mulgval 13458 mulgfng 13460 subgmulg 13524 gsumfzreidx 13673 gsumfzsubmcl 13674 gsumfzmptfidmadd 13675 gsumfzmhm 13679 gsumfzfsum 14350 lgsval 15481 lgscllem 15484 lgsneg 15501 lgsdilem 15504 lgsdir 15512 lgsdi 15514 lgsne0 15515 lgsquadlemsfi 15552 lgsquadlem3 15556

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