zabsle1 - Intuitionistic Logic Explorer

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Theorem zabsle1 16032

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 3752	. . 3 ⊢ (𝑍 ∈ {-1, 0, 1} → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
2		fveq2 5690	. . . . 5 ⊢ (𝑍 = -1 → (abs‘𝑍) = (abs‘-1))
3		ax-1cn 8262	. . . . . . . 8 ⊢ 1 ∈ ℂ
4	3	absnegi 11891	. . . . . . 7 ⊢ (abs‘-1) = (abs‘1)
5		abs1 11816	. . . . . . 7 ⊢ (abs‘1) = 1
6	4, 5	eqtri 2259	. . . . . 6 ⊢ (abs‘-1) = 1
7		1le1 8890	. . . . . 6 ⊢ 1 ≤ 1
8	6, 7	eqbrtri 4146	. . . . 5 ⊢ (abs‘-1) ≤ 1
9	2, 8	eqbrtrdi 4164	. . . 4 ⊢ (𝑍 = -1 → (abs‘𝑍) ≤ 1)
10		fveq2 5690	. . . . 5 ⊢ (𝑍 = 0 → (abs‘𝑍) = (abs‘0))
11		abs0 11802	. . . . . 6 ⊢ (abs‘0) = 0
12		0le1 8799	. . . . . 6 ⊢ 0 ≤ 1
13	11, 12	eqbrtri 4146	. . . . 5 ⊢ (abs‘0) ≤ 1
14	10, 13	eqbrtrdi 4164	. . . 4 ⊢ (𝑍 = 0 → (abs‘𝑍) ≤ 1)
15		fveq2 5690	. . . . 5 ⊢ (𝑍 = 1 → (abs‘𝑍) = (abs‘1))
16	5, 7	eqbrtri 4146	. . . . 5 ⊢ (abs‘1) ≤ 1
17	15, 16	eqbrtrdi 4164	. . . 4 ⊢ (𝑍 = 1 → (abs‘𝑍) ≤ 1)
18	9, 14, 17	3jaoi 1344	. . 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 9627	. . . 4 ⊢ (𝑍 ∈ ℤ → 𝑍 ∈ ℝ)
21		1red 8331	. . . 4 ⊢ (𝑍 ∈ ℤ → 1 ∈ ℝ)
22	20, 21	absled 11919	. . 3 ⊢ (𝑍 ∈ ℤ → ((abs‘𝑍) ≤ 1 ↔ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)))
23		elz 9625	. . . 4 ⊢ (𝑍 ∈ ℤ ↔ (𝑍 ∈ ℝ ∧ (𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ)))
24		3mix2 1198	. . . . . . . . . 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 9307	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ ℕ → (𝑍 ≤ 1 ↔ 𝑍 = 1))
27	26	biimpac 298	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ≤ 1 ∧ 𝑍 ∈ ℕ) → 𝑍 = 1)
28	27	3mix3d 1205	. . . . . . . . . . . . 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 9646	. . . . . . . . . 10 ⊢ (-𝑍 ∈ ℕ ↔ (-𝑍 ∈ ℤ ∧ 1 ≤ -𝑍))
34		1red 8331	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ ℝ → 1 ∈ ℝ)
35		lenegcon2 8785	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ 𝑍 ∈ ℝ) → (1 ≤ -𝑍 ↔ 𝑍 ≤ -1))
36	34, 35	mpancom 426	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ ℝ → (1 ≤ -𝑍 ↔ 𝑍 ≤ -1))
37		neg1rr 9389	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ∈ ℝ
38	37	a1i 9	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 ∈ ℝ → -1 ∈ ℝ)
39		id 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 ∈ ℝ → 𝑍 ∈ ℝ)
40	38, 39	letri3d 8431	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑍 ∈ ℝ → (-1 = 𝑍 ↔ (-1 ≤ 𝑍 ∧ 𝑍 ≤ -1)))
41		3mix1 1197	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 = -1 → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
42	41	eqcoms 2241	. . . . . . . . . . . . . . . . . . 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 1344	. . . . . . . 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 3750	. . . . . . . . 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 1008 = wceq 1402 ∈ wcel 2209 {ctp 3707 class class class wbr 4125 ‘cfv 5372 ℝcr 8168 0cc0 8169 1c1 8170 ≤ cle 8351 -cneg 8488 ℕcn 9283 ℤcz 9623 abscabs 11741

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-tp 3713 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-frec 6652 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-n0 9543 df-z 9624 df-uz 9901 df-rp 10034 df-seqfrec 10863 df-exp 10954 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743

This theorem is referenced by: lgscl1 16056

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