Theorem fz0to4untppr 10458

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 9297	. . . . 5 ⊢ 3 = (2 + 1)
2		2cn 9308	. . . . . . . 8 ⊢ 2 ∈ ℂ
3	2	addlidi 8416	. . . . . . 7 ⊢ (0 + 2) = 2
4	3	eqcomi 2236	. . . . . 6 ⊢ 2 = (0 + 2)
5	4	oveq1i 6060	. . . . 5 ⊢ (2 + 1) = ((0 + 2) + 1)
6	1, 5	eqtri 2253	. . . 4 ⊢ 3 = ((0 + 2) + 1)
7		3z 9606	. . . . 5 ⊢ 3 ∈ ℤ
8		0re 8274	. . . . . 6 ⊢ 0 ∈ ℝ
9		3re 9311	. . . . . 6 ⊢ 3 ∈ ℝ
10		3pos 9331	. . . . . 6 ⊢ 0 < 3
11	8, 9, 10	ltleii 8376	. . . . 5 ⊢ 0 ≤ 3
12		0z 9588	. . . . . 6 ⊢ 0 ∈ ℤ
13	12	eluz1i 9861	. . . . 5 ⊢ (3 ∈ (ℤ≥‘0) ↔ (3 ∈ ℤ ∧ 0 ≤ 3))
14	7, 11, 13	mpbir2an 951	. . . 4 ⊢ 3 ∈ (ℤ≥‘0)
15	6, 14	eqeltrri 2306	. . 3 ⊢ ((0 + 2) + 1) ∈ (ℤ≥‘0)
16		4z 9607	. . . . 5 ⊢ 4 ∈ ℤ
17		2re 9307	. . . . . 6 ⊢ 2 ∈ ℝ
18		4re 9314	. . . . . 6 ⊢ 4 ∈ ℝ
19		2lt4 9411	. . . . . 6 ⊢ 2 < 4
20	17, 18, 19	ltleii 8376	. . . . 5 ⊢ 2 ≤ 4
21		2z 9605	. . . . . 6 ⊢ 2 ∈ ℤ
22	21	eluz1i 9861	. . . . 5 ⊢ (4 ∈ (ℤ≥‘2) ↔ (4 ∈ ℤ ∧ 2 ≤ 4))
23	16, 20, 22	mpbir2an 951	. . . 4 ⊢ 4 ∈ (ℤ≥‘2)
24	4	fveq2i 5673	. . . 4 ⊢ (ℤ≥‘2) = (ℤ≥‘(0 + 2))
25	23, 24	eleqtri 2307	. . 3 ⊢ 4 ∈ (ℤ≥‘(0 + 2))
26		fzsplit2 10384	. . 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 10412	. . . . 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 8220	. . . . 5 ⊢ 1 ∈ ℂ
31		eqidd 2233	. . . . . 6 ⊢ (1 ∈ ℂ → 0 = 0)
32		addlid 8412	. . . . . 6 ⊢ (1 ∈ ℂ → (0 + 1) = 1)
33	3	a1i 9	. . . . . 6 ⊢ (1 ∈ ℂ → (0 + 2) = 2)
34	31, 32, 33	tpeq123d 3783	. . . . 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 2253	. . 3 ⊢ (0...(0 + 2)) = {0, 1, 2}
37	3	a1i 9	. . . . . . . 8 ⊢ (3 ∈ ℤ → (0 + 2) = 2)
38	37	oveq1d 6065	. . . . . . 7 ⊢ (3 ∈ ℤ → ((0 + 2) + 1) = (2 + 1))
39	38, 1	eqtr4di 2283	. . . . . 6 ⊢ (3 ∈ ℤ → ((0 + 2) + 1) = 3)
40	39	oveq1d 6065	. . . . 5 ⊢ (3 ∈ ℤ → (((0 + 2) + 1)...4) = (3...4))
41		eqid 2232	. . . . . . . . . 10 ⊢ 3 = 3
42		df-4 9298	. . . . . . . . . 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 9410	. . . . . . . . . . 11 ⊢ 3 < 4
46	9, 18, 45	ltleii 8376	. . . . . . . . . 10 ⊢ 3 ≤ 4
47	7	eluz1i 9861	. . . . . . . . . 10 ⊢ (4 ∈ (ℤ≥‘3) ↔ (4 ∈ ℤ ∧ 3 ≤ 4))
48	16, 46, 47	mpbir2an 951	. . . . . . . . 9 ⊢ 4 ∈ (ℤ≥‘3)
49		fzopth 10395	. . . . . . . . 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 10411	. . . . . . 7 ⊢ (3 ∈ ℤ → (3...(3 + 1)) = {3, (3 + 1)})
53	51, 52	eqtrd 2265	. . . . . 6 ⊢ (3 ∈ ℤ → (3...4) = {3, (3 + 1)})
54	42	eqcomi 2236	. . . . . . 7 ⊢ (3 + 1) = 4
55	54	preq2i 3772	. . . . . 6 ⊢ {3, (3 + 1)} = {3, 4}
56	53, 55	eqtrdi 2281	. . . . 5 ⊢ (3 ∈ ℤ → (3...4) = {3, 4})
57	40, 56	eqtrd 2265	. . . 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 3371	. 2 ⊢ ((0...(0 + 2)) ∪ (((0 + 2) + 1)...4)) = ({0, 1, 2} ∪ {3, 4})
60	27, 59	eqtri 2253	1 ⊢ (0...4) = ({0, 1, 2} ∪ {3, 4})

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203   ∪ cun 3209  {cpr 3690  {ctp 3691   class class class wbr 4109  ‘cfv 5352  (class class class)co 6050  ℂcc 8125  0cc0 8127  1c1 8128   + caddc 8130   ≤ cle 8309  2c2 9288  3c3 9289  4c4 9290  ℤcz 9577  ℤ_≥cuz 9853  ...cfz 10342

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-tp 3697  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343

This theorem is referenced by:  prm23lt5  12961