Theorem fz0to4untppr 10349

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 9193	. . . . 5 ⊢ 3 = (2 + 1)
2		2cn 9204	. . . . . . . 8 ⊢ 2 ∈ ℂ
3	2	addlidi 8312	. . . . . . 7 ⊢ (0 + 2) = 2
4	3	eqcomi 2233	. . . . . 6 ⊢ 2 = (0 + 2)
5	4	oveq1i 6023	. . . . 5 ⊢ (2 + 1) = ((0 + 2) + 1)
6	1, 5	eqtri 2250	. . . 4 ⊢ 3 = ((0 + 2) + 1)
7		3z 9498	. . . . 5 ⊢ 3 ∈ ℤ
8		0re 8169	. . . . . 6 ⊢ 0 ∈ ℝ
9		3re 9207	. . . . . 6 ⊢ 3 ∈ ℝ
10		3pos 9227	. . . . . 6 ⊢ 0 < 3
11	8, 9, 10	ltleii 8272	. . . . 5 ⊢ 0 ≤ 3
12		0z 9480	. . . . . 6 ⊢ 0 ∈ ℤ
13	12	eluz1i 9753	. . . . 5 ⊢ (3 ∈ (ℤ≥‘0) ↔ (3 ∈ ℤ ∧ 0 ≤ 3))
14	7, 11, 13	mpbir2an 948	. . . 4 ⊢ 3 ∈ (ℤ≥‘0)
15	6, 14	eqeltrri 2303	. . 3 ⊢ ((0 + 2) + 1) ∈ (ℤ≥‘0)
16		4z 9499	. . . . 5 ⊢ 4 ∈ ℤ
17		2re 9203	. . . . . 6 ⊢ 2 ∈ ℝ
18		4re 9210	. . . . . 6 ⊢ 4 ∈ ℝ
19		2lt4 9307	. . . . . 6 ⊢ 2 < 4
20	17, 18, 19	ltleii 8272	. . . . 5 ⊢ 2 ≤ 4
21		2z 9497	. . . . . 6 ⊢ 2 ∈ ℤ
22	21	eluz1i 9753	. . . . 5 ⊢ (4 ∈ (ℤ≥‘2) ↔ (4 ∈ ℤ ∧ 2 ≤ 4))
23	16, 20, 22	mpbir2an 948	. . . 4 ⊢ 4 ∈ (ℤ≥‘2)
24	4	fveq2i 5638	. . . 4 ⊢ (ℤ≥‘2) = (ℤ≥‘(0 + 2))
25	23, 24	eleqtri 2304	. . 3 ⊢ 4 ∈ (ℤ≥‘(0 + 2))
26		fzsplit2 10275	. . 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 10303	. . . . 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 8115	. . . . 5 ⊢ 1 ∈ ℂ
31		eqidd 2230	. . . . . 6 ⊢ (1 ∈ ℂ → 0 = 0)
32		addlid 8308	. . . . . 6 ⊢ (1 ∈ ℂ → (0 + 1) = 1)
33	3	a1i 9	. . . . . 6 ⊢ (1 ∈ ℂ → (0 + 2) = 2)
34	31, 32, 33	tpeq123d 3761	. . . . 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 2250	. . 3 ⊢ (0...(0 + 2)) = {0, 1, 2}
37	3	a1i 9	. . . . . . . 8 ⊢ (3 ∈ ℤ → (0 + 2) = 2)
38	37	oveq1d 6028	. . . . . . 7 ⊢ (3 ∈ ℤ → ((0 + 2) + 1) = (2 + 1))
39	38, 1	eqtr4di 2280	. . . . . 6 ⊢ (3 ∈ ℤ → ((0 + 2) + 1) = 3)
40	39	oveq1d 6028	. . . . 5 ⊢ (3 ∈ ℤ → (((0 + 2) + 1)...4) = (3...4))
41		eqid 2229	. . . . . . . . . 10 ⊢ 3 = 3
42		df-4 9194	. . . . . . . . . 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 9306	. . . . . . . . . . 11 ⊢ 3 < 4
46	9, 18, 45	ltleii 8272	. . . . . . . . . 10 ⊢ 3 ≤ 4
47	7	eluz1i 9753	. . . . . . . . . 10 ⊢ (4 ∈ (ℤ≥‘3) ↔ (4 ∈ ℤ ∧ 3 ≤ 4))
48	16, 46, 47	mpbir2an 948	. . . . . . . . 9 ⊢ 4 ∈ (ℤ≥‘3)
49		fzopth 10286	. . . . . . . . 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 10302	. . . . . . 7 ⊢ (3 ∈ ℤ → (3...(3 + 1)) = {3, (3 + 1)})
53	51, 52	eqtrd 2262	. . . . . 6 ⊢ (3 ∈ ℤ → (3...4) = {3, (3 + 1)})
54	42	eqcomi 2233	. . . . . . 7 ⊢ (3 + 1) = 4
55	54	preq2i 3750	. . . . . 6 ⊢ {3, (3 + 1)} = {3, 4}
56	53, 55	eqtrdi 2278	. . . . 5 ⊢ (3 ∈ ℤ → (3...4) = {3, 4})
57	40, 56	eqtrd 2262	. . . 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 3357	. 2 ⊢ ((0...(0 + 2)) ∪ (((0 + 2) + 1)...4)) = ({0, 1, 2} ∪ {3, 4})
60	27, 59	eqtri 2250	1 ⊢ (0...4) = ({0, 1, 2} ∪ {3, 4})

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200   ∪ cun 3196  {cpr 3668  {ctp 3669   class class class wbr 4086  ‘cfv 5324  (class class class)co 6013  ℂcc 8020  0cc0 8022  1c1 8023   + caddc 8025   ≤ cle 8205  2c2 9184  3c3 9185  4c4 9186  ℤcz 9469  ℤ_≥cuz 9745  ...cfz 10233

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-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  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-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-tp 3675  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234

This theorem is referenced by:  prm23lt5  12826