zabsle1 - Intuitionistic Logic Explorer

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Theorem zabsle1 15693

Description: {-1, 0, 1} is the set of all integers with absolute value at most 1. (Contributed by AV, 13-Jul-2021.)

**Assertion**
Ref	Expression
zabsle1	⊢ (𝑍 ∈ ℤ → (𝑍 ∈ {-1, 0, 1} ↔ (abs‘𝑍) ≤ 1))

**Proof of Theorem zabsle1**
Step	Hyp	Ref	Expression
1		eltpi 3713	. . 3 ⊢ (𝑍 ∈ {-1, 0, 1} → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
2		fveq2 5629	. . . . 5 ⊢ (𝑍 = -1 → (abs‘𝑍) = (abs‘-1))
3		ax-1cn 8103	. . . . . . . 8 ⊢ 1 ∈ ℂ
4	3	absnegi 11673	. . . . . . 7 ⊢ (abs‘-1) = (abs‘1)
5		abs1 11598	. . . . . . 7 ⊢ (abs‘1) = 1
6	4, 5	eqtri 2250	. . . . . 6 ⊢ (abs‘-1) = 1
7		1le1 8730	. . . . . 6 ⊢ 1 ≤ 1
8	6, 7	eqbrtri 4104	. . . . 5 ⊢ (abs‘-1) ≤ 1
9	2, 8	eqbrtrdi 4122	. . . 4 ⊢ (𝑍 = -1 → (abs‘𝑍) ≤ 1)
10		fveq2 5629	. . . . 5 ⊢ (𝑍 = 0 → (abs‘𝑍) = (abs‘0))
11		abs0 11584	. . . . . 6 ⊢ (abs‘0) = 0
12		0le1 8639	. . . . . 6 ⊢ 0 ≤ 1
13	11, 12	eqbrtri 4104	. . . . 5 ⊢ (abs‘0) ≤ 1
14	10, 13	eqbrtrdi 4122	. . . 4 ⊢ (𝑍 = 0 → (abs‘𝑍) ≤ 1)
15		fveq2 5629	. . . . 5 ⊢ (𝑍 = 1 → (abs‘𝑍) = (abs‘1))
16	5, 7	eqbrtri 4104	. . . . 5 ⊢ (abs‘1) ≤ 1
17	15, 16	eqbrtrdi 4122	. . . 4 ⊢ (𝑍 = 1 → (abs‘𝑍) ≤ 1)
18	9, 14, 17	3jaoi 1337	. . 3 ⊢ ((𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1) → (abs‘𝑍) ≤ 1)
19	1, 18	syl 14	. 2 ⊢ (𝑍 ∈ {-1, 0, 1} → (abs‘𝑍) ≤ 1)
20		zre 9461	. . . 4 ⊢ (𝑍 ∈ ℤ → 𝑍 ∈ ℝ)
21		1red 8172	. . . 4 ⊢ (𝑍 ∈ ℤ → 1 ∈ ℝ)
22	20, 21	absled 11701	. . 3 ⊢ (𝑍 ∈ ℤ → ((abs‘𝑍) ≤ 1 ↔ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)))
23		elz 9459	. . . 4 ⊢ (𝑍 ∈ ℤ ↔ (𝑍 ∈ ℝ ∧ (𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ)))
24		3mix2 1191	. . . . . . . . . 10 ⊢ (𝑍 = 0 → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
25	24	a1d 22	. . . . . . . . 9 ⊢ (𝑍 = 0 → ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
26		nnle1eq1 9145	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ ℕ → (𝑍 ≤ 1 ↔ 𝑍 = 1))
27	26	biimpac 298	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ≤ 1 ∧ 𝑍 ∈ ℕ) → 𝑍 = 1)
28	27	3mix3d 1198	. . . . . . . . . . . . 13 ⊢ ((𝑍 ≤ 1 ∧ 𝑍 ∈ ℕ) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
29	28	ex 115	. . . . . . . . . . . 12 ⊢ (𝑍 ≤ 1 → (𝑍 ∈ ℕ → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
30	29	adantl 277	. . . . . . . . . . 11 ⊢ ((-1 ≤ 𝑍 ∧ 𝑍 ≤ 1) → (𝑍 ∈ ℕ → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
31	30	adantl 277	. . . . . . . . . 10 ⊢ ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 ∈ ℕ → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
32	31	com12 30	. . . . . . . . 9 ⊢ (𝑍 ∈ ℕ → ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
33		elnnz1 9480	. . . . . . . . . 10 ⊢ (-𝑍 ∈ ℕ ↔ (-𝑍 ∈ ℤ ∧ 1 ≤ -𝑍))
34		1red 8172	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ ℝ → 1 ∈ ℝ)
35		lenegcon2 8625	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ 𝑍 ∈ ℝ) → (1 ≤ -𝑍 ↔ 𝑍 ≤ -1))
36	34, 35	mpancom 422	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ ℝ → (1 ≤ -𝑍 ↔ 𝑍 ≤ -1))
37		neg1rr 9227	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ∈ ℝ
38	37	a1i 9	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 ∈ ℝ → -1 ∈ ℝ)
39		id 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 ∈ ℝ → 𝑍 ∈ ℝ)
40	38, 39	letri3d 8273	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑍 ∈ ℝ → (-1 = 𝑍 ↔ (-1 ≤ 𝑍 ∧ 𝑍 ≤ -1)))
41		3mix1 1190	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 = -1 → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
42	41	eqcoms 2232	. . . . . . . . . . . . . . . . . . 19 ⊢ (-1 = 𝑍 → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
43	40, 42	biimtrrdi 164	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑍 ∈ ℝ → ((-1 ≤ 𝑍 ∧ 𝑍 ≤ -1) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
44	43	com12 30	. . . . . . . . . . . . . . . . 17 ⊢ ((-1 ≤ 𝑍 ∧ 𝑍 ≤ -1) → (𝑍 ∈ ℝ → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
45	44	ex 115	. . . . . . . . . . . . . . . 16 ⊢ (-1 ≤ 𝑍 → (𝑍 ≤ -1 → (𝑍 ∈ ℝ → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))))
46	45	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((-1 ≤ 𝑍 ∧ 𝑍 ≤ 1) → (𝑍 ≤ -1 → (𝑍 ∈ ℝ → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))))
47	46	com13 80	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ ℝ → (𝑍 ≤ -1 → ((-1 ≤ 𝑍 ∧ 𝑍 ≤ 1) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))))
48	36, 47	sylbid 150	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ ℝ → (1 ≤ -𝑍 → ((-1 ≤ 𝑍 ∧ 𝑍 ≤ 1) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))))
49	48	com12 30	. . . . . . . . . . . 12 ⊢ (1 ≤ -𝑍 → (𝑍 ∈ ℝ → ((-1 ≤ 𝑍 ∧ 𝑍 ≤ 1) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))))
50	49	impd 254	. . . . . . . . . . 11 ⊢ (1 ≤ -𝑍 → ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
51	50	adantl 277	. . . . . . . . . 10 ⊢ ((-𝑍 ∈ ℤ ∧ 1 ≤ -𝑍) → ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
52	33, 51	sylbi 121	. . . . . . . . 9 ⊢ (-𝑍 ∈ ℕ → ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
53	25, 32, 52	3jaoi 1337	. . . . . . . 8 ⊢ ((𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ) → ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
54	53	imp 124	. . . . . . 7 ⊢ (((𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ) ∧ (𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1))) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
55		eltpg 3711	. . . . . . . . 9 ⊢ (𝑍 ∈ ℝ → (𝑍 ∈ {-1, 0, 1} ↔ (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
56	55	adantr 276	. . . . . . . 8 ⊢ ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 ∈ {-1, 0, 1} ↔ (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
57	56	adantl 277	. . . . . . 7 ⊢ (((𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ) ∧ (𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1))) → (𝑍 ∈ {-1, 0, 1} ↔ (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
58	54, 57	mpbird 167	. . . . . 6 ⊢ (((𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ) ∧ (𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1))) → 𝑍 ∈ {-1, 0, 1})
59	58	exp32 365	. . . . 5 ⊢ ((𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ) → (𝑍 ∈ ℝ → ((-1 ≤ 𝑍 ∧ 𝑍 ≤ 1) → 𝑍 ∈ {-1, 0, 1})))
60	59	impcom 125	. . . 4 ⊢ ((𝑍 ∈ ℝ ∧ (𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ)) → ((-1 ≤ 𝑍 ∧ 𝑍 ≤ 1) → 𝑍 ∈ {-1, 0, 1}))
61	23, 60	sylbi 121	. . 3 ⊢ (𝑍 ∈ ℤ → ((-1 ≤ 𝑍 ∧ 𝑍 ≤ 1) → 𝑍 ∈ {-1, 0, 1}))
62	22, 61	sylbid 150	. 2 ⊢ (𝑍 ∈ ℤ → ((abs‘𝑍) ≤ 1 → 𝑍 ∈ {-1, 0, 1}))
63	19, 62	impbid2 143	1 ⊢ (𝑍 ∈ ℤ → (𝑍 ∈ {-1, 0, 1} ↔ (abs‘𝑍) ≤ 1))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ w3o 1001 = wceq 1395 ∈ wcel 2200 {ctp 3668 class class class wbr 4083 ‘cfv 5318 ℝcr 8009 0cc0 8010 1c1 8011 ≤ cle 8193 -cneg 8329 ℕcn 9121 ℤcz 9457 abscabs 11523

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-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130

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-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-tp 3674 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-n0 9381 df-z 9458 df-uz 9734 df-rp 9862 df-seqfrec 10682 df-exp 10773 df-cj 11368 df-re 11369 df-im 11370 df-rsqrt 11524 df-abs 11525

This theorem is referenced by: lgscl1 15717

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