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Metamath Proof Explorer

Theorem fzo0to42pr

Description: A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017)

Ref Expression
Assertion fzo0to42pr 
|- ( 0 ..^ 4 ) = ( { 0 , 1 } u. { 2 , 3 } )

Proof

Step Hyp Ref Expression
1 2nn0 
 |-  2 e. NN0
2 4nn0 
 |-  4 e. NN0
3 2re 
 |-  2 e. RR
4 4re 
 |-  4 e. RR
5 2lt4 
 |-  2 < 4
6 3 4 5 ltleii 
 |-  2 <_ 4
7 elfz2nn0 
 |-  ( 2 e. ( 0 ... 4 ) <-> ( 2 e. NN0 /\ 4 e. NN0 /\ 2 <_ 4 ) )
8 1 2 6 7 mpbir3an 
 |-  2 e. ( 0 ... 4 )
9 fzosplit 
 |-  ( 2 e. ( 0 ... 4 ) -> ( 0 ..^ 4 ) = ( ( 0 ..^ 2 ) u. ( 2 ..^ 4 ) ) )
10 8 9 ax-mp 
 |-  ( 0 ..^ 4 ) = ( ( 0 ..^ 2 ) u. ( 2 ..^ 4 ) )
11 fzo0to2pr 
 |-  ( 0 ..^ 2 ) = { 0 , 1 }
12 4z 
 |-  4 e. ZZ
13 fzoval 
 |-  ( 4 e. ZZ -> ( 2 ..^ 4 ) = ( 2 ... ( 4 - 1 ) ) )
14 12 13 ax-mp 
 |-  ( 2 ..^ 4 ) = ( 2 ... ( 4 - 1 ) )
15 4m1e3 
 |-  ( 4 - 1 ) = 3
16 df-3 
 |-  3 = ( 2 + 1 )
17 15 16 eqtri 
 |-  ( 4 - 1 ) = ( 2 + 1 )
18 17 oveq2i 
 |-  ( 2 ... ( 4 - 1 ) ) = ( 2 ... ( 2 + 1 ) )
19 2z 
 |-  2 e. ZZ
20 fzpr 
 |-  ( 2 e. ZZ -> ( 2 ... ( 2 + 1 ) ) = { 2 , ( 2 + 1 ) } )
21 19 20 ax-mp 
 |-  ( 2 ... ( 2 + 1 ) ) = { 2 , ( 2 + 1 ) }
22 18 21 eqtri 
 |-  ( 2 ... ( 4 - 1 ) ) = { 2 , ( 2 + 1 ) }
23 2p1e3 
 |-  ( 2 + 1 ) = 3
24 23 preq2i 
 |-  { 2 , ( 2 + 1 ) } = { 2 , 3 }
25 14 22 24 3eqtri 
 |-  ( 2 ..^ 4 ) = { 2 , 3 }
26 11 25 uneq12i 
 |-  ( ( 0 ..^ 2 ) u. ( 2 ..^ 4 ) ) = ( { 0 , 1 } u. { 2 , 3 } )
27 10 26 eqtri 
 |-  ( 0 ..^ 4 ) = ( { 0 , 1 } u. { 2 , 3 } )