Theorem fz0to4untppr 10421

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 9262	. . . . 5 |- 3 = ( 2 + 1 )
2		2cn 9273	. . . . . . . 8 |- 2 e. CC
3	2	addlidi 8381	. . . . . . 7 |- ( 0 + 2 ) = 2
4	3	eqcomi 2235	. . . . . 6 |- 2 = ( 0 + 2 )
5	4	oveq1i 6038	. . . . 5 |- ( 2 + 1 ) = ( ( 0 + 2 ) + 1 )
6	1, 5	eqtri 2252	. . . 4 |- 3 = ( ( 0 + 2 ) + 1 )
7		3z 9569	. . . . 5 |- 3 e. ZZ
8		0re 8239	. . . . . 6 |- 0 e. RR
9		3re 9276	. . . . . 6 |- 3 e. RR
10		3pos 9296	. . . . . 6 |- 0 < 3
11	8, 9, 10	ltleii 8341	. . . . 5 |- 0 <_ 3
12		0z 9551	. . . . . 6 |- 0 e. ZZ
13	12	eluz1i 9824	. . . . 5 |- ( 3 e. ( ZZ>= ` 0 ) <-> ( 3 e. ZZ /\ 0 <_ 3 ) )
14	7, 11, 13	mpbir2an 951	. . . 4 |- 3 e. ( ZZ>= ` 0 )
15	6, 14	eqeltrri 2305	. . 3 |- ( ( 0 + 2 ) + 1 ) e. ( ZZ>= ` 0 )
16		4z 9570	. . . . 5 |- 4 e. ZZ
17		2re 9272	. . . . . 6 |- 2 e. RR
18		4re 9279	. . . . . 6 |- 4 e. RR
19		2lt4 9376	. . . . . 6 |- 2 < 4
20	17, 18, 19	ltleii 8341	. . . . 5 |- 2 <_ 4
21		2z 9568	. . . . . 6 |- 2 e. ZZ
22	21	eluz1i 9824	. . . . 5 |- ( 4 e. ( ZZ>= ` 2 ) <-> ( 4 e. ZZ /\ 2 <_ 4 ) )
23	16, 20, 22	mpbir2an 951	. . . 4 |- 4 e. ( ZZ>= ` 2 )
24	4	fveq2i 5651	. . . 4 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 0 + 2 ) )
25	23, 24	eleqtri 2306	. . 3 |- 4 e. ( ZZ>= ` ( 0 + 2 ) )
26		fzsplit2 10347	. . 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 426	. 2 |- ( 0 ... 4 ) = ( ( 0 ... ( 0 + 2 ) ) u. ( ( ( 0 + 2 ) + 1 ) ... 4 ) )
28		fztp 10375	. . . . 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 8185	. . . . 5 |- 1 e. CC
31		eqidd 2232	. . . . . 6 |- ( 1 e. CC -> 0 = 0 )
32		addlid 8377	. . . . . 6 |- ( 1 e. CC -> ( 0 + 1 ) = 1 )
33	3	a1i 9	. . . . . 6 |- ( 1 e. CC -> ( 0 + 2 ) = 2 )
34	31, 32, 33	tpeq123d 3767	. . . . 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 2252	. . 3 |- ( 0 ... ( 0 + 2 ) ) = { 0 , 1 , 2 }
37	3	a1i 9	. . . . . . . 8 |- ( 3 e. ZZ -> ( 0 + 2 ) = 2 )
38	37	oveq1d 6043	. . . . . . 7 |- ( 3 e. ZZ -> ( ( 0 + 2 ) + 1 ) = ( 2 + 1 ) )
39	38, 1	eqtr4di 2282	. . . . . 6 |- ( 3 e. ZZ -> ( ( 0 + 2 ) + 1 ) = 3 )
40	39	oveq1d 6043	. . . . 5 |- ( 3 e. ZZ -> ( ( ( 0 + 2 ) + 1 ) ... 4 ) = ( 3 ... 4 ) )
41		eqid 2231	. . . . . . . . . 10 |- 3 = 3
42		df-4 9263	. . . . . . . . . 10 |- 4 = ( 3 + 1 )
43	41, 42	pm3.2i 272	. . . . . . . . 9 |- ( 3 = 3 /\ 4 = ( 3 + 1 ) )
44	43	a1i 9	. . . . . . . 8 |- ( 3 e. ZZ -> ( 3 = 3 /\ 4 = ( 3 + 1 ) ) )
45		3lt4 9375	. . . . . . . . . . 11 |- 3 < 4
46	9, 18, 45	ltleii 8341	. . . . . . . . . 10 |- 3 <_ 4
47	7	eluz1i 9824	. . . . . . . . . 10 |- ( 4 e. ( ZZ>= ` 3 ) <-> ( 4 e. ZZ /\ 3 <_ 4 ) )
48	16, 46, 47	mpbir2an 951	. . . . . . . . 9 |- 4 e. ( ZZ>= ` 3 )
49		fzopth 10358	. . . . . . . . 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 134	. . . . . . 7 |- ( 3 e. ZZ -> ( 3 ... 4 ) = ( 3 ... ( 3 + 1 ) ) )
52		fzpr 10374	. . . . . . 7 |- ( 3 e. ZZ -> ( 3 ... ( 3 + 1 ) ) = { 3 , ( 3 + 1 ) } )
53	51, 52	eqtrd 2264	. . . . . 6 |- ( 3 e. ZZ -> ( 3 ... 4 ) = { 3 , ( 3 + 1 ) } )
54	42	eqcomi 2235	. . . . . . 7 |- ( 3 + 1 ) = 4
55	54	preq2i 3756	. . . . . 6 |- { 3 , ( 3 + 1 ) } = { 3 , 4 }
56	53, 55	eqtrdi 2280	. . . . 5 |- ( 3 e. ZZ -> ( 3 ... 4 ) = { 3 , 4 } )
57	40, 56	eqtrd 2264	. . . 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 3361	. 2 |- ( ( 0 ... ( 0 + 2 ) ) u. ( ( ( 0 + 2 ) + 1 ) ... 4 ) ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )
60	27, 59	eqtri 2252	1 |- ( 0 ... 4 ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   u. cun 3199   {cpr 3674   {ctp 3675   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8090   0cc0 8092   1c1 8093   + caddc 8095   <_ cle 8274   2c2 9253   3c3 9254   4c4 9255   ZZcz 9540   ZZ>=cuz 9816   ...cfz 10305

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-fz 10306

This theorem is referenced by:  prm23lt5  12916