Theorem fz0to4untppr 10281

Description: An integer range from 0 to 4 is the union of a triple and a pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)

Assertion

Ref	Expression
fz0to4untppr	⊢ (0...4) = ({0, 1, 2} ∪ {3, 4})

Proof of Theorem fz0to4untppr

Step	Hyp	Ref	Expression
1		df-3 9131	. . . . 5 ⊢ 3 = (2 + 1)
2		2cn 9142	. . . . . . . 8 ⊢ 2 ∈ ℂ
3	2	addlidi 8250	. . . . . . 7 ⊢ (0 + 2) = 2
4	3	eqcomi 2211	. . . . . 6 ⊢ 2 = (0 + 2)
5	4	oveq1i 5977	. . . . 5 ⊢ (2 + 1) = ((0 + 2) + 1)
6	1, 5	eqtri 2228	. . . 4 ⊢ 3 = ((0 + 2) + 1)
7		3z 9436	. . . . 5 ⊢ 3 ∈ ℤ
8		0re 8107	. . . . . 6 ⊢ 0 ∈ ℝ
9		3re 9145	. . . . . 6 ⊢ 3 ∈ ℝ
10		3pos 9165	. . . . . 6 ⊢ 0 < 3
11	8, 9, 10	ltleii 8210	. . . . 5 ⊢ 0 ≤ 3
12		0z 9418	. . . . . 6 ⊢ 0 ∈ ℤ
13	12	eluz1i 9690	. . . . 5 ⊢ (3 ∈ (ℤ≥‘0) ↔ (3 ∈ ℤ ∧ 0 ≤ 3))
14	7, 11, 13	mpbir2an 945	. . . 4 ⊢ 3 ∈ (ℤ≥‘0)
15	6, 14	eqeltrri 2281	. . 3 ⊢ ((0 + 2) + 1) ∈ (ℤ≥‘0)
16		4z 9437	. . . . 5 ⊢ 4 ∈ ℤ
17		2re 9141	. . . . . 6 ⊢ 2 ∈ ℝ
18		4re 9148	. . . . . 6 ⊢ 4 ∈ ℝ
19		2lt4 9245	. . . . . 6 ⊢ 2 < 4
20	17, 18, 19	ltleii 8210	. . . . 5 ⊢ 2 ≤ 4
21		2z 9435	. . . . . 6 ⊢ 2 ∈ ℤ
22	21	eluz1i 9690	. . . . 5 ⊢ (4 ∈ (ℤ≥‘2) ↔ (4 ∈ ℤ ∧ 2 ≤ 4))
23	16, 20, 22	mpbir2an 945	. . . 4 ⊢ 4 ∈ (ℤ≥‘2)
24	4	fveq2i 5602	. . . 4 ⊢ (ℤ≥‘2) = (ℤ≥‘(0 + 2))
25	23, 24	eleqtri 2282	. . 3 ⊢ 4 ∈ (ℤ≥‘(0 + 2))
26		fzsplit2 10207	. . 3 ⊢ ((((0 + 2) + 1) ∈ (ℤ≥‘0) ∧ 4 ∈ (ℤ≥‘(0 + 2))) → (0...4) = ((0...(0 + 2)) ∪ (((0 + 2) + 1)...4)))
27	15, 25, 26	mp2an 426	. 2 ⊢ (0...4) = ((0...(0 + 2)) ∪ (((0 + 2) + 1)...4))
28		fztp 10235	. . . . 5 ⊢ (0 ∈ ℤ → (0...(0 + 2)) = {0, (0 + 1), (0 + 2)})
29	12, 28	ax-mp 5	. . . 4 ⊢ (0...(0 + 2)) = {0, (0 + 1), (0 + 2)}
30		ax-1cn 8053	. . . . 5 ⊢ 1 ∈ ℂ
31		eqidd 2208	. . . . . 6 ⊢ (1 ∈ ℂ → 0 = 0)
32		addlid 8246	. . . . . 6 ⊢ (1 ∈ ℂ → (0 + 1) = 1)
33	3	a1i 9	. . . . . 6 ⊢ (1 ∈ ℂ → (0 + 2) = 2)
34	31, 32, 33	tpeq123d 3735	. . . . 5 ⊢ (1 ∈ ℂ → {0, (0 + 1), (0 + 2)} = {0, 1, 2})
35	30, 34	ax-mp 5	. . . 4 ⊢ {0, (0 + 1), (0 + 2)} = {0, 1, 2}
36	29, 35	eqtri 2228	. . 3 ⊢ (0...(0 + 2)) = {0, 1, 2}
37	3	a1i 9	. . . . . . . 8 ⊢ (3 ∈ ℤ → (0 + 2) = 2)
38	37	oveq1d 5982	. . . . . . 7 ⊢ (3 ∈ ℤ → ((0 + 2) + 1) = (2 + 1))
39	38, 1	eqtr4di 2258	. . . . . 6 ⊢ (3 ∈ ℤ → ((0 + 2) + 1) = 3)
40	39	oveq1d 5982	. . . . 5 ⊢ (3 ∈ ℤ → (((0 + 2) + 1)...4) = (3...4))
41		eqid 2207	. . . . . . . . . 10 ⊢ 3 = 3
42		df-4 9132	. . . . . . . . . 10 ⊢ 4 = (3 + 1)
43	41, 42	pm3.2i 272	. . . . . . . . 9 ⊢ (3 = 3 ∧ 4 = (3 + 1))
44	43	a1i 9	. . . . . . . 8 ⊢ (3 ∈ ℤ → (3 = 3 ∧ 4 = (3 + 1)))
45		3lt4 9244	. . . . . . . . . . 11 ⊢ 3 < 4
46	9, 18, 45	ltleii 8210	. . . . . . . . . 10 ⊢ 3 ≤ 4
47	7	eluz1i 9690	. . . . . . . . . 10 ⊢ (4 ∈ (ℤ≥‘3) ↔ (4 ∈ ℤ ∧ 3 ≤ 4))
48	16, 46, 47	mpbir2an 945	. . . . . . . . 9 ⊢ 4 ∈ (ℤ≥‘3)
49		fzopth 10218	. . . . . . . . 9 ⊢ (4 ∈ (ℤ≥‘3) → ((3...4) = (3...(3 + 1)) ↔ (3 = 3 ∧ 4 = (3 + 1))))
50	48, 49	ax-mp 5	. . . . . . . 8 ⊢ ((3...4) = (3...(3 + 1)) ↔ (3 = 3 ∧ 4 = (3 + 1)))
51	44, 50	sylibr 134	. . . . . . 7 ⊢ (3 ∈ ℤ → (3...4) = (3...(3 + 1)))
52		fzpr 10234	. . . . . . 7 ⊢ (3 ∈ ℤ → (3...(3 + 1)) = {3, (3 + 1)})
53	51, 52	eqtrd 2240	. . . . . 6 ⊢ (3 ∈ ℤ → (3...4) = {3, (3 + 1)})
54	42	eqcomi 2211	. . . . . . 7 ⊢ (3 + 1) = 4
55	54	preq2i 3724	. . . . . 6 ⊢ {3, (3 + 1)} = {3, 4}
56	53, 55	eqtrdi 2256	. . . . 5 ⊢ (3 ∈ ℤ → (3...4) = {3, 4})
57	40, 56	eqtrd 2240	. . . 4 ⊢ (3 ∈ ℤ → (((0 + 2) + 1)...4) = {3, 4})
58	7, 57	ax-mp 5	. . 3 ⊢ (((0 + 2) + 1)...4) = {3, 4}
59	36, 58	uneq12i 3333	. 2 ⊢ ((0...(0 + 2)) ∪ (((0 + 2) + 1)...4)) = ({0, 1, 2} ∪ {3, 4})
60	27, 59	eqtri 2228	1 ⊢ (0...4) = ({0, 1, 2} ∪ {3, 4})

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2178   ∪ cun 3172  {cpr 3644  {ctp 3645   class class class wbr 4059  ‘cfv 5290  (class class class)co 5967  ℂcc 7958  0cc0 7960  1c1 7961   + caddc 7963   ≤ cle 8143  2c2 9122  3c3 9123  4c4 9124  ℤcz 9407  ℤ_≥cuz 9683  ...cfz 10165

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166

This theorem is referenced by:  prm23lt5  12701