Theorem fz0to4untppr 10320

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 9170	. . . . 5 ⊢ 3 = (2 + 1)
2		2cn 9181	. . . . . . . 8 ⊢ 2 ∈ ℂ
3	2	addlidi 8289	. . . . . . 7 ⊢ (0 + 2) = 2
4	3	eqcomi 2233	. . . . . 6 ⊢ 2 = (0 + 2)
5	4	oveq1i 6011	. . . . 5 ⊢ (2 + 1) = ((0 + 2) + 1)
6	1, 5	eqtri 2250	. . . 4 ⊢ 3 = ((0 + 2) + 1)
7		3z 9475	. . . . 5 ⊢ 3 ∈ ℤ
8		0re 8146	. . . . . 6 ⊢ 0 ∈ ℝ
9		3re 9184	. . . . . 6 ⊢ 3 ∈ ℝ
10		3pos 9204	. . . . . 6 ⊢ 0 < 3
11	8, 9, 10	ltleii 8249	. . . . 5 ⊢ 0 ≤ 3
12		0z 9457	. . . . . 6 ⊢ 0 ∈ ℤ
13	12	eluz1i 9729	. . . . 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 9476	. . . . 5 ⊢ 4 ∈ ℤ
17		2re 9180	. . . . . 6 ⊢ 2 ∈ ℝ
18		4re 9187	. . . . . 6 ⊢ 4 ∈ ℝ
19		2lt4 9284	. . . . . 6 ⊢ 2 < 4
20	17, 18, 19	ltleii 8249	. . . . 5 ⊢ 2 ≤ 4
21		2z 9474	. . . . . 6 ⊢ 2 ∈ ℤ
22	21	eluz1i 9729	. . . . 5 ⊢ (4 ∈ (ℤ≥‘2) ↔ (4 ∈ ℤ ∧ 2 ≤ 4))
23	16, 20, 22	mpbir2an 948	. . . 4 ⊢ 4 ∈ (ℤ≥‘2)
24	4	fveq2i 5630	. . . 4 ⊢ (ℤ≥‘2) = (ℤ≥‘(0 + 2))
25	23, 24	eleqtri 2304	. . 3 ⊢ 4 ∈ (ℤ≥‘(0 + 2))
26		fzsplit2 10246	. . 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 10274	. . . . 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 8092	. . . . 5 ⊢ 1 ∈ ℂ
31		eqidd 2230	. . . . . 6 ⊢ (1 ∈ ℂ → 0 = 0)
32		addlid 8285	. . . . . 6 ⊢ (1 ∈ ℂ → (0 + 1) = 1)
33	3	a1i 9	. . . . . 6 ⊢ (1 ∈ ℂ → (0 + 2) = 2)
34	31, 32, 33	tpeq123d 3758	. . . . 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 6016	. . . . . . 7 ⊢ (3 ∈ ℤ → ((0 + 2) + 1) = (2 + 1))
39	38, 1	eqtr4di 2280	. . . . . 6 ⊢ (3 ∈ ℤ → ((0 + 2) + 1) = 3)
40	39	oveq1d 6016	. . . . 5 ⊢ (3 ∈ ℤ → (((0 + 2) + 1)...4) = (3...4))
41		eqid 2229	. . . . . . . . . 10 ⊢ 3 = 3
42		df-4 9171	. . . . . . . . . 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 9283	. . . . . . . . . . 11 ⊢ 3 < 4
46	9, 18, 45	ltleii 8249	. . . . . . . . . 10 ⊢ 3 ≤ 4
47	7	eluz1i 9729	. . . . . . . . . 10 ⊢ (4 ∈ (ℤ≥‘3) ↔ (4 ∈ ℤ ∧ 3 ≤ 4))
48	16, 46, 47	mpbir2an 948	. . . . . . . . 9 ⊢ 4 ∈ (ℤ≥‘3)
49		fzopth 10257	. . . . . . . . 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 10273	. . . . . . 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 3747	. . . . . 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 3356	. 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 3195  {cpr 3667  {ctp 3668   class class class wbr 4083  ‘cfv 5318  (class class class)co 6001  ℂcc 7997  0cc0 7999  1c1 8000   + caddc 8002   ≤ cle 8182  2c2 9161  3c3 9162  4c4 9163  ℤcz 9446  ℤ_≥cuz 9722  ...cfz 10204

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115

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 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-fz 10205

This theorem is referenced by:  prm23lt5  12786