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Metamath Proof Explorer

Theorem fzo1to4tp

Description: A half-open integer range from 1 to 4 is an unordered triple. (Contributed by AV, 28-Jul-2021)

Ref Expression
Assertion fzo1to4tp 
|- ( 1 ..^ 4 ) = { 1 , 2 , 3 }

Proof

Step Hyp Ref Expression
1 4z 
 |-  4 e. ZZ
2 fzoval 
 |-  ( 4 e. ZZ -> ( 1 ..^ 4 ) = ( 1 ... ( 4 - 1 ) ) )
3 1 2 ax-mp 
 |-  ( 1 ..^ 4 ) = ( 1 ... ( 4 - 1 ) )
4 4m1e3 
 |-  ( 4 - 1 ) = 3
5 df-3 
 |-  3 = ( 2 + 1 )
6 2cn 
 |-  2 e. CC
7 ax-1cn 
 |-  1 e. CC
8 6 7 addcomi 
 |-  ( 2 + 1 ) = ( 1 + 2 )
9 4 5 8 3eqtri 
 |-  ( 4 - 1 ) = ( 1 + 2 )
10 9 oveq2i 
 |-  ( 1 ... ( 4 - 1 ) ) = ( 1 ... ( 1 + 2 ) )
11 1z 
 |-  1 e. ZZ
12 fztp 
 |-  ( 1 e. ZZ -> ( 1 ... ( 1 + 2 ) ) = { 1 , ( 1 + 1 ) , ( 1 + 2 ) } )
13 eqidd 
 |-  ( 1 e. ZZ -> 1 = 1 )
14 1p1e2 
 |-  ( 1 + 1 ) = 2
15 14 a1i 
 |-  ( 1 e. ZZ -> ( 1 + 1 ) = 2 )
16 1p2e3 
 |-  ( 1 + 2 ) = 3
17 16 a1i 
 |-  ( 1 e. ZZ -> ( 1 + 2 ) = 3 )
18 13 15 17 tpeq123d 
 |-  ( 1 e. ZZ -> { 1 , ( 1 + 1 ) , ( 1 + 2 ) } = { 1 , 2 , 3 } )
19 12 18 eqtrd 
 |-  ( 1 e. ZZ -> ( 1 ... ( 1 + 2 ) ) = { 1 , 2 , 3 } )
20 11 19 ax-mp 
 |-  ( 1 ... ( 1 + 2 ) ) = { 1 , 2 , 3 }
21 3 10 20 3eqtri 
 |-  ( 1 ..^ 4 ) = { 1 , 2 , 3 }