Theorem fz0to4untppr 10358

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 9202	. . . . 5 |- 3 = ( 2 + 1 )
2		2cn 9213	. . . . . . . 8 |- 2 e. CC
3	2	addlidi 8321	. . . . . . 7 |- ( 0 + 2 ) = 2
4	3	eqcomi 2235	. . . . . 6 |- 2 = ( 0 + 2 )
5	4	oveq1i 6027	. . . . 5 |- ( 2 + 1 ) = ( ( 0 + 2 ) + 1 )
6	1, 5	eqtri 2252	. . . 4 |- 3 = ( ( 0 + 2 ) + 1 )
7		3z 9507	. . . . 5 |- 3 e. ZZ
8		0re 8178	. . . . . 6 |- 0 e. RR
9		3re 9216	. . . . . 6 |- 3 e. RR
10		3pos 9236	. . . . . 6 |- 0 < 3
11	8, 9, 10	ltleii 8281	. . . . 5 |- 0 <_ 3
12		0z 9489	. . . . . 6 |- 0 e. ZZ
13	12	eluz1i 9762	. . . . 5 |- ( 3 e. ( ZZ>= ` 0 ) <-> ( 3 e. ZZ /\ 0 <_ 3 ) )
14	7, 11, 13	mpbir2an 950	. . . 4 |- 3 e. ( ZZ>= ` 0 )
15	6, 14	eqeltrri 2305	. . 3 |- ( ( 0 + 2 ) + 1 ) e. ( ZZ>= ` 0 )
16		4z 9508	. . . . 5 |- 4 e. ZZ
17		2re 9212	. . . . . 6 |- 2 e. RR
18		4re 9219	. . . . . 6 |- 4 e. RR
19		2lt4 9316	. . . . . 6 |- 2 < 4
20	17, 18, 19	ltleii 8281	. . . . 5 |- 2 <_ 4
21		2z 9506	. . . . . 6 |- 2 e. ZZ
22	21	eluz1i 9762	. . . . 5 |- ( 4 e. ( ZZ>= ` 2 ) <-> ( 4 e. ZZ /\ 2 <_ 4 ) )
23	16, 20, 22	mpbir2an 950	. . . 4 |- 4 e. ( ZZ>= ` 2 )
24	4	fveq2i 5642	. . . 4 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 0 + 2 ) )
25	23, 24	eleqtri 2306	. . 3 |- 4 e. ( ZZ>= ` ( 0 + 2 ) )
26		fzsplit2 10284	. . 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 10312	. . . . 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 8124	. . . . 5 |- 1 e. CC
31		eqidd 2232	. . . . . 6 |- ( 1 e. CC -> 0 = 0 )
32		addlid 8317	. . . . . 6 |- ( 1 e. CC -> ( 0 + 1 ) = 1 )
33	3	a1i 9	. . . . . 6 |- ( 1 e. CC -> ( 0 + 2 ) = 2 )
34	31, 32, 33	tpeq123d 3763	. . . . 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 6032	. . . . . . 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 6032	. . . . 5 |- ( 3 e. ZZ -> ( ( ( 0 + 2 ) + 1 ) ... 4 ) = ( 3 ... 4 ) )
41		eqid 2231	. . . . . . . . . 10 |- 3 = 3
42		df-4 9203	. . . . . . . . . 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 9315	. . . . . . . . . . 11 |- 3 < 4
46	9, 18, 45	ltleii 8281	. . . . . . . . . 10 |- 3 <_ 4
47	7	eluz1i 9762	. . . . . . . . . 10 |- ( 4 e. ( ZZ>= ` 3 ) <-> ( 4 e. ZZ /\ 3 <_ 4 ) )
48	16, 46, 47	mpbir2an 950	. . . . . . . . 9 |- 4 e. ( ZZ>= ` 3 )
49		fzopth 10295	. . . . . . . . 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 10311	. . . . . . 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 3752	. . . . . 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 3359	. 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 1397   e. wcel 2202   u. cun 3198   {cpr 3670   {ctp 3671   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   + caddc 8034   <_ cle 8214   2c2 9193   3c3 9194   4c4 9195   ZZcz 9478   ZZ>=cuz 9754   ...cfz 10242

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243

This theorem is referenced by:  prm23lt5  12835