Theorem fz0to4untppr 10509

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 9343	. . . . 5 ⊢ 3 = (2 + 1)
2		2cn 9354	. . . . . . . 8 ⊢ 2 ∈ ℂ
3	2	addlidi 8459	. . . . . . 7 ⊢ (0 + 2) = 2
4	3	eqcomi 2242	. . . . . 6 ⊢ 2 = (0 + 2)
5	4	oveq1i 6085	. . . . 5 ⊢ (2 + 1) = ((0 + 2) + 1)
6	1, 5	eqtri 2259	. . . 4 ⊢ 3 = ((0 + 2) + 1)
7		3z 9652	. . . . 5 ⊢ 3 ∈ ℤ
8		0re 8316	. . . . . 6 ⊢ 0 ∈ ℝ
9		3re 9357	. . . . . 6 ⊢ 3 ∈ ℝ
10		3pos 9377	. . . . . 6 ⊢ 0 < 3
11	8, 9, 10	ltleii 8418	. . . . 5 ⊢ 0 ≤ 3
12		0z 9634	. . . . . 6 ⊢ 0 ∈ ℤ
13	12	eluz1i 9908	. . . . 5 ⊢ (3 ∈ (ℤ≥‘0) ↔ (3 ∈ ℤ ∧ 0 ≤ 3))
14	7, 11, 13	mpbir2an 955	. . . 4 ⊢ 3 ∈ (ℤ≥‘0)
15	6, 14	eqeltrri 2312	. . 3 ⊢ ((0 + 2) + 1) ∈ (ℤ≥‘0)
16		4z 9653	. . . . 5 ⊢ 4 ∈ ℤ
17		2re 9353	. . . . . 6 ⊢ 2 ∈ ℝ
18		4re 9360	. . . . . 6 ⊢ 4 ∈ ℝ
19		2lt4 9457	. . . . . 6 ⊢ 2 < 4
20	17, 18, 19	ltleii 8418	. . . . 5 ⊢ 2 ≤ 4
21		2z 9651	. . . . . 6 ⊢ 2 ∈ ℤ
22	21	eluz1i 9908	. . . . 5 ⊢ (4 ∈ (ℤ≥‘2) ↔ (4 ∈ ℤ ∧ 2 ≤ 4))
23	16, 20, 22	mpbir2an 955	. . . 4 ⊢ 4 ∈ (ℤ≥‘2)
24	4	fveq2i 5693	. . . 4 ⊢ (ℤ≥‘2) = (ℤ≥‘(0 + 2))
25	23, 24	eleqtri 2313	. . 3 ⊢ 4 ∈ (ℤ≥‘(0 + 2))
26		fzsplit2 10433	. . 3 ⊢ ((((0 + 2) + 1) ∈ (ℤ≥‘0) ∧ 4 ∈ (ℤ≥‘(0 + 2))) → (0...4) = ((0...(0 + 2)) ∪ (((0 + 2) + 1)...4)))
27	15, 25, 26	mp2an 430	. 2 ⊢ (0...4) = ((0...(0 + 2)) ∪ (((0 + 2) + 1)...4))
28		fztp 10463	. . . . 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 8262	. . . . 5 ⊢ 1 ∈ ℂ
31		eqidd 2239	. . . . . 6 ⊢ (1 ∈ ℂ → 0 = 0)
32		addlid 8455	. . . . . 6 ⊢ (1 ∈ ℂ → (0 + 1) = 1)
33	3	a1i 9	. . . . . 6 ⊢ (1 ∈ ℂ → (0 + 2) = 2)
34	31, 32, 33	tpeq123d 3799	. . . . 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 2259	. . 3 ⊢ (0...(0 + 2)) = {0, 1, 2}
37	3	a1i 9	. . . . . . . 8 ⊢ (3 ∈ ℤ → (0 + 2) = 2)
38	37	oveq1d 6090	. . . . . . 7 ⊢ (3 ∈ ℤ → ((0 + 2) + 1) = (2 + 1))
39	38, 1	eqtr4di 2289	. . . . . 6 ⊢ (3 ∈ ℤ → ((0 + 2) + 1) = 3)
40	39	oveq1d 6090	. . . . 5 ⊢ (3 ∈ ℤ → (((0 + 2) + 1)...4) = (3...4))
41		eqid 2238	. . . . . . . . . 10 ⊢ 3 = 3
42		df-4 9344	. . . . . . . . . 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 9456	. . . . . . . . . . 11 ⊢ 3 < 4
46	9, 18, 45	ltleii 8418	. . . . . . . . . 10 ⊢ 3 ≤ 4
47	7	eluz1i 9908	. . . . . . . . . 10 ⊢ (4 ∈ (ℤ≥‘3) ↔ (4 ∈ ℤ ∧ 3 ≤ 4))
48	16, 46, 47	mpbir2an 955	. . . . . . . . 9 ⊢ 4 ∈ (ℤ≥‘3)
49		fzopth 10445	. . . . . . . . 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 10462	. . . . . . 7 ⊢ (3 ∈ ℤ → (3...(3 + 1)) = {3, (3 + 1)})
53	51, 52	eqtrd 2271	. . . . . 6 ⊢ (3 ∈ ℤ → (3...4) = {3, (3 + 1)})
54	42	eqcomi 2242	. . . . . . 7 ⊢ (3 + 1) = 4
55	54	preq2i 3788	. . . . . 6 ⊢ {3, (3 + 1)} = {3, 4}
56	53, 55	eqtrdi 2287	. . . . 5 ⊢ (3 ∈ ℤ → (3...4) = {3, 4})
57	40, 56	eqtrd 2271	. . . 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 3381	. 2 ⊢ ((0...(0 + 2)) ∪ (((0 + 2) + 1)...4)) = ({0, 1, 2} ∪ {3, 4})
60	27, 59	eqtri 2259	1 ⊢ (0...4) = ({0, 1, 2} ∪ {3, 4})

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209   ∪ cun 3218  {cpr 3706  {ctp 3707   class class class wbr 4125  ‘cfv 5372  (class class class)co 6075  ℂcc 8167  0cc0 8169  1c1 8170   + caddc 8172   ≤ cle 8351  2c2 9334  3c3 9335  4c4 9336  ℤcz 9623  ℤ_≥cuz 9900  ...cfz 10390

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-tp 3713  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391

This theorem is referenced by:  prm23lt5  13020