zabsle1 - Intuitionistic Logic Explorer

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Theorem zabsle1 13694

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 3630	. . 3 ⊢ (𝑍 ∈ {-1, 0, 1} → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
2		fveq2 5496	. . . . 5 ⊢ (𝑍 = -1 → (abs‘𝑍) = (abs‘-1))
3		ax-1cn 7867	. . . . . . . 8 ⊢ 1 ∈ ℂ
4	3	absnegi 11111	. . . . . . 7 ⊢ (abs‘-1) = (abs‘1)
5		abs1 11036	. . . . . . 7 ⊢ (abs‘1) = 1
6	4, 5	eqtri 2191	. . . . . 6 ⊢ (abs‘-1) = 1
7		1le1 8491	. . . . . 6 ⊢ 1 ≤ 1
8	6, 7	eqbrtri 4010	. . . . 5 ⊢ (abs‘-1) ≤ 1
9	2, 8	eqbrtrdi 4028	. . . 4 ⊢ (𝑍 = -1 → (abs‘𝑍) ≤ 1)
10		fveq2 5496	. . . . 5 ⊢ (𝑍 = 0 → (abs‘𝑍) = (abs‘0))
11		abs0 11022	. . . . . 6 ⊢ (abs‘0) = 0
12		0le1 8400	. . . . . 6 ⊢ 0 ≤ 1
13	11, 12	eqbrtri 4010	. . . . 5 ⊢ (abs‘0) ≤ 1
14	10, 13	eqbrtrdi 4028	. . . 4 ⊢ (𝑍 = 0 → (abs‘𝑍) ≤ 1)
15		fveq2 5496	. . . . 5 ⊢ (𝑍 = 1 → (abs‘𝑍) = (abs‘1))
16	5, 7	eqbrtri 4010	. . . . 5 ⊢ (abs‘1) ≤ 1
17	15, 16	eqbrtrdi 4028	. . . 4 ⊢ (𝑍 = 1 → (abs‘𝑍) ≤ 1)
18	9, 14, 17	3jaoi 1298	. . 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 9216	. . . 4 ⊢ (𝑍 ∈ ℤ → 𝑍 ∈ ℝ)
21		1red 7935	. . . 4 ⊢ (𝑍 ∈ ℤ → 1 ∈ ℝ)
22	20, 21	absled 11139	. . 3 ⊢ (𝑍 ∈ ℤ → ((abs‘𝑍) ≤ 1 ↔ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)))
23		elz 9214	. . . 4 ⊢ (𝑍 ∈ ℤ ↔ (𝑍 ∈ ℝ ∧ (𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ)))
24		3mix2 1162	. . . . . . . . . 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 8902	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ ℕ → (𝑍 ≤ 1 ↔ 𝑍 = 1))
27	26	biimpac 296	. . . . . . . . . . . . . 14 ⊢ ((𝑍 ≤ 1 ∧ 𝑍 ∈ ℕ) → 𝑍 = 1)
28	27	3mix3d 1169	. . . . . . . . . . . . 13 ⊢ ((𝑍 ≤ 1 ∧ 𝑍 ∈ ℕ) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
29	28	ex 114	. . . . . . . . . . . 12 ⊢ (𝑍 ≤ 1 → (𝑍 ∈ ℕ → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
30	29	adantl 275	. . . . . . . . . . 11 ⊢ ((-1 ≤ 𝑍 ∧ 𝑍 ≤ 1) → (𝑍 ∈ ℕ → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
31	30	adantl 275	. . . . . . . . . 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 9235	. . . . . . . . . 10 ⊢ (-𝑍 ∈ ℕ ↔ (-𝑍 ∈ ℤ ∧ 1 ≤ -𝑍))
34		1red 7935	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ ℝ → 1 ∈ ℝ)
35		lenegcon2 8386	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ 𝑍 ∈ ℝ) → (1 ≤ -𝑍 ↔ 𝑍 ≤ -1))
36	34, 35	mpancom 420	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ ℝ → (1 ≤ -𝑍 ↔ 𝑍 ≤ -1))
37		neg1rr 8984	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ∈ ℝ
38	37	a1i 9	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 ∈ ℝ → -1 ∈ ℝ)
39		id 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 ∈ ℝ → 𝑍 ∈ ℝ)
40	38, 39	letri3d 8035	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑍 ∈ ℝ → (-1 = 𝑍 ↔ (-1 ≤ 𝑍 ∧ 𝑍 ≤ -1)))
41		3mix1 1161	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 = -1 → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
42	41	eqcoms 2173	. . . . . . . . . . . . . . . . . . 19 ⊢ (-1 = 𝑍 → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
43	40, 42	syl6bir 163	. . . . . . . . . . . . . . . . . 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 114	. . . . . . . . . . . . . . . 16 ⊢ (-1 ≤ 𝑍 → (𝑍 ≤ -1 → (𝑍 ∈ ℝ → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))))
46	45	adantr 274	. . . . . . . . . . . . . . 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 149	. . . . . . . . . . . . 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 252	. . . . . . . . . . 11 ⊢ (1 ≤ -𝑍 → ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
51	50	adantl 275	. . . . . . . . . 10 ⊢ ((-𝑍 ∈ ℤ ∧ 1 ≤ -𝑍) → ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
52	33, 51	sylbi 120	. . . . . . . . 9 ⊢ (-𝑍 ∈ ℕ → ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
53	25, 32, 52	3jaoi 1298	. . . . . . . 8 ⊢ ((𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ) → ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
54	53	imp 123	. . . . . . 7 ⊢ (((𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ) ∧ (𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1))) → (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1))
55		eltpg 3628	. . . . . . . . 9 ⊢ (𝑍 ∈ ℝ → (𝑍 ∈ {-1, 0, 1} ↔ (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
56	55	adantr 274	. . . . . . . 8 ⊢ ((𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1)) → (𝑍 ∈ {-1, 0, 1} ↔ (𝑍 = -1 ∨ 𝑍 = 0 ∨ 𝑍 = 1)))
57	56	adantl 275	. . . . . . 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 166	. . . . . 6 ⊢ (((𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ) ∧ (𝑍 ∈ ℝ ∧ (-1 ≤ 𝑍 ∧ 𝑍 ≤ 1))) → 𝑍 ∈ {-1, 0, 1})
59	58	exp32 363	. . . . 5 ⊢ ((𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ) → (𝑍 ∈ ℝ → ((-1 ≤ 𝑍 ∧ 𝑍 ≤ 1) → 𝑍 ∈ {-1, 0, 1})))
60	59	impcom 124	. . . 4 ⊢ ((𝑍 ∈ ℝ ∧ (𝑍 = 0 ∨ 𝑍 ∈ ℕ ∨ -𝑍 ∈ ℕ)) → ((-1 ≤ 𝑍 ∧ 𝑍 ≤ 1) → 𝑍 ∈ {-1, 0, 1}))
61	23, 60	sylbi 120	. . 3 ⊢ (𝑍 ∈ ℤ → ((-1 ≤ 𝑍 ∧ 𝑍 ≤ 1) → 𝑍 ∈ {-1, 0, 1}))
62	22, 61	sylbid 149	. 2 ⊢ (𝑍 ∈ ℤ → ((abs‘𝑍) ≤ 1 → 𝑍 ∈ {-1, 0, 1}))
63	19, 62	impbid2 142	1 ⊢ (𝑍 ∈ ℤ → (𝑍 ∈ {-1, 0, 1} ↔ (abs‘𝑍) ≤ 1))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ w3o 972 = wceq 1348 ∈ wcel 2141 {ctp 3585 class class class wbr 3989 ‘cfv 5198 ℝcr 7773 0cc0 7774 1c1 7775 ≤ cle 7955 -cneg 8091 ℕcn 8878 ℤcz 9212 abscabs 10961

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-tp 3591 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-rp 9611 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963

This theorem is referenced by: lgscl1 13718

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