Theorem fz0to4untppr 10080

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 } u. { 3 , 4 } )

Proof of Theorem fz0to4untppr

Step	Hyp	Ref	Expression
1		df-3 8938	. . . . 5 |- 3 = ( 2 + 1 )
2		2cn 8949	. . . . . . . 8 |- 2 e. CC
3	2	addid2i 8062	. . . . . . 7 |- ( 0 + 2 ) = 2
4	3	eqcomi 2174	. . . . . 6 |- 2 = ( 0 + 2 )
5	4	oveq1i 5863	. . . . 5 |- ( 2 + 1 ) = ( ( 0 + 2 ) + 1 )
6	1, 5	eqtri 2191	. . . 4 |- 3 = ( ( 0 + 2 ) + 1 )
7		3z 9241	. . . . 5 |- 3 e. ZZ
8		0re 7920	. . . . . 6 |- 0 e. RR
9		3re 8952	. . . . . 6 |- 3 e. RR
10		3pos 8972	. . . . . 6 |- 0 < 3
11	8, 9, 10	ltleii 8022	. . . . 5 |- 0 <_ 3
12		0z 9223	. . . . . 6 |- 0 e. ZZ
13	12	eluz1i 9494	. . . . 5 |- ( 3 e. ( ZZ>= ` 0 ) <-> ( 3 e. ZZ /\ 0 <_ 3 ) )
14	7, 11, 13	mpbir2an 937	. . . 4 |- 3 e. ( ZZ>= ` 0 )
15	6, 14	eqeltrri 2244	. . 3 |- ( ( 0 + 2 ) + 1 ) e. ( ZZ>= ` 0 )
16		4z 9242	. . . . 5 |- 4 e. ZZ
17		2re 8948	. . . . . 6 |- 2 e. RR
18		4re 8955	. . . . . 6 |- 4 e. RR
19		2lt4 9051	. . . . . 6 |- 2 < 4
20	17, 18, 19	ltleii 8022	. . . . 5 |- 2 <_ 4
21		2z 9240	. . . . . 6 |- 2 e. ZZ
22	21	eluz1i 9494	. . . . 5 |- ( 4 e. ( ZZ>= ` 2 ) <-> ( 4 e. ZZ /\ 2 <_ 4 ) )
23	16, 20, 22	mpbir2an 937	. . . 4 |- 4 e. ( ZZ>= ` 2 )
24	4	fveq2i 5499	. . . 4 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 0 + 2 ) )
25	23, 24	eleqtri 2245	. . 3 |- 4 e. ( ZZ>= ` ( 0 + 2 ) )
26		fzsplit2 10006	. . 3 |- ( ( ( ( 0 + 2 ) + 1 ) e. ( ZZ>= ` 0 ) /\ 4 e. ( ZZ>= ` ( 0 + 2 ) ) ) -> ( 0 ... 4 ) = ( ( 0 ... ( 0 + 2 ) ) u. ( ( ( 0 + 2 ) + 1 ) ... 4 ) ) )
27	15, 25, 26	mp2an 424	. 2 |- ( 0 ... 4 ) = ( ( 0 ... ( 0 + 2 ) ) u. ( ( ( 0 + 2 ) + 1 ) ... 4 ) )
28		fztp 10034	. . . . 5 |- ( 0 e. ZZ -> ( 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 7867	. . . . 5 |- 1 e. CC
31		eqidd 2171	. . . . . 6 |- ( 1 e. CC -> 0 = 0 )
32		addid2 8058	. . . . . 6 |- ( 1 e. CC -> ( 0 + 1 ) = 1 )
33	3	a1i 9	. . . . . 6 |- ( 1 e. CC -> ( 0 + 2 ) = 2 )
34	31, 32, 33	tpeq123d 3675	. . . . 5 |- ( 1 e. CC -> { 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 2191	. . 3 |- ( 0 ... ( 0 + 2 ) ) = { 0 , 1 , 2 }
37	3	a1i 9	. . . . . . . 8 |- ( 3 e. ZZ -> ( 0 + 2 ) = 2 )
38	37	oveq1d 5868	. . . . . . 7 |- ( 3 e. ZZ -> ( ( 0 + 2 ) + 1 ) = ( 2 + 1 ) )
39	38, 1	eqtr4di 2221	. . . . . 6 |- ( 3 e. ZZ -> ( ( 0 + 2 ) + 1 ) = 3 )
40	39	oveq1d 5868	. . . . 5 |- ( 3 e. ZZ -> ( ( ( 0 + 2 ) + 1 ) ... 4 ) = ( 3 ... 4 ) )
41		eqid 2170	. . . . . . . . . 10 |- 3 = 3
42		df-4 8939	. . . . . . . . . 10 |- 4 = ( 3 + 1 )
43	41, 42	pm3.2i 270	. . . . . . . . 9 |- ( 3 = 3 /\ 4 = ( 3 + 1 ) )
44	43	a1i 9	. . . . . . . 8 |- ( 3 e. ZZ -> ( 3 = 3 /\ 4 = ( 3 + 1 ) ) )
45		3lt4 9050	. . . . . . . . . . 11 |- 3 < 4
46	9, 18, 45	ltleii 8022	. . . . . . . . . 10 |- 3 <_ 4
47	7	eluz1i 9494	. . . . . . . . . 10 |- ( 4 e. ( ZZ>= ` 3 ) <-> ( 4 e. ZZ /\ 3 <_ 4 ) )
48	16, 46, 47	mpbir2an 937	. . . . . . . . 9 |- 4 e. ( ZZ>= ` 3 )
49		fzopth 10017	. . . . . . . . 9 |- ( 4 e. ( ZZ>= ` 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 133	. . . . . . 7 |- ( 3 e. ZZ -> ( 3 ... 4 ) = ( 3 ... ( 3 + 1 ) ) )
52		fzpr 10033	. . . . . . 7 |- ( 3 e. ZZ -> ( 3 ... ( 3 + 1 ) ) = { 3 , ( 3 + 1 ) } )
53	51, 52	eqtrd 2203	. . . . . 6 |- ( 3 e. ZZ -> ( 3 ... 4 ) = { 3 , ( 3 + 1 ) } )
54	42	eqcomi 2174	. . . . . . 7 |- ( 3 + 1 ) = 4
55	54	preq2i 3664	. . . . . 6 |- { 3 , ( 3 + 1 ) } = { 3 , 4 }
56	53, 55	eqtrdi 2219	. . . . 5 |- ( 3 e. ZZ -> ( 3 ... 4 ) = { 3 , 4 } )
57	40, 56	eqtrd 2203	. . . 4 |- ( 3 e. ZZ -> ( ( ( 0 + 2 ) + 1 ) ... 4 ) = { 3 , 4 } )
58	7, 57	ax-mp 5	. . 3 |- ( ( ( 0 + 2 ) + 1 ) ... 4 ) = { 3 , 4 }
59	36, 58	uneq12i 3279	. 2 |- ( ( 0 ... ( 0 + 2 ) ) u. ( ( ( 0 + 2 ) + 1 ) ... 4 ) ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )
60	27, 59	eqtri 2191	1 |- ( 0 ... 4 ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   u. cun 3119   {cpr 3584   {ctp 3585   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   <_ cle 7955   2c2 8929   3c3 8930   4c4 8931   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-tp 3591  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966

This theorem is referenced by:  prm23lt5  12217