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Metamath Proof Explorer

Theorem fz0to4untppr

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) (Proof shortened by AV, 7-Aug-2025)

Ref Expression
Assertion fz0to4untppr 
|- ( 0 ... 4 ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )

Proof

Step Hyp Ref Expression
1 2p1e3 
 |-  ( 2 + 1 ) = 3
2 0z 
 |-  0 e. ZZ
3 3z 
 |-  3 e. ZZ
4 0re 
 |-  0 e. RR
5 3re 
 |-  3 e. RR
6 3pos 
 |-  0 < 3
7 4 5 6 ltleii 
 |-  0 <_ 3
8 eluz2 
 |-  ( 3 e. ( ZZ>= ` 0 ) <-> ( 0 e. ZZ /\ 3 e. ZZ /\ 0 <_ 3 ) )
9 2 3 7 8 mpbir3an 
 |-  3 e. ( ZZ>= ` 0 )
10 1 9 eqeltri 
 |-  ( 2 + 1 ) e. ( ZZ>= ` 0 )
11 2z 
 |-  2 e. ZZ
12 4z 
 |-  4 e. ZZ
13 2re 
 |-  2 e. RR
14 4re 
 |-  4 e. RR
15 2lt4 
 |-  2 < 4
16 13 14 15 ltleii 
 |-  2 <_ 4
17 eluz2 
 |-  ( 4 e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ 4 e. ZZ /\ 2 <_ 4 ) )
18 11 12 16 17 mpbir3an 
 |-  4 e. ( ZZ>= ` 2 )
19 fzsplit2 
 |-  ( ( ( 2 + 1 ) e. ( ZZ>= ` 0 ) /\ 4 e. ( ZZ>= ` 2 ) ) -> ( 0 ... 4 ) = ( ( 0 ... 2 ) u. ( ( 2 + 1 ) ... 4 ) ) )
20 10 18 19 mp2an 
 |-  ( 0 ... 4 ) = ( ( 0 ... 2 ) u. ( ( 2 + 1 ) ... 4 ) )
21 fz0tp 
 |-  ( 0 ... 2 ) = { 0 , 1 , 2 }
22 1 oveq1i 
 |-  ( ( 2 + 1 ) ... 4 ) = ( 3 ... 4 )
23 df-4 
 |-  4 = ( 3 + 1 )
24 23 oveq2i 
 |-  ( 3 ... 4 ) = ( 3 ... ( 3 + 1 ) )
25 fzpr 
 |-  ( 3 e. ZZ -> ( 3 ... ( 3 + 1 ) ) = { 3 , ( 3 + 1 ) } )
26 3 25 ax-mp 
 |-  ( 3 ... ( 3 + 1 ) ) = { 3 , ( 3 + 1 ) }
27 3p1e4 
 |-  ( 3 + 1 ) = 4
28 27 preq2i 
 |-  { 3 , ( 3 + 1 ) } = { 3 , 4 }
29 24 26 28 3eqtri 
 |-  ( 3 ... 4 ) = { 3 , 4 }
30 22 29 eqtri 
 |-  ( ( 2 + 1 ) ... 4 ) = { 3 , 4 }
31 21 30 uneq12i 
 |-  ( ( 0 ... 2 ) u. ( ( 2 + 1 ) ... 4 ) ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )
32 20 31 eqtri 
 |-  ( 0 ... 4 ) = ( { 0 , 1 , 2 } u. { 3 , 4 } )