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Metamath Proof Explorer

Theorem ex-ceil

Description: Example for df-ceil . (Contributed by AV, 4-Sep-2021)

Ref Expression
Assertion ex-ceil ( ( ⌈ ‘ ( 3 / 2 ) ) = 2 ∧ ( ⌈ ‘ - ( 3 / 2 ) ) = - 1 )

Proof

Step Hyp Ref Expression
1 ex-fl ⊢ ( ( ⌊ ‘ ( 3 / 2 ) ) = 1 ∧ ( ⌊ ‘ - ( 3 / 2 ) ) = - 2 )
2 3re ⊢ 3 ∈ ℝ
3 2 rehalfcli ⊢ ( 3 / 2 ) ∈ ℝ
4 3 renegcli ⊢ - ( 3 / 2 ) ∈ ℝ
5 ceilval ⊢ ( - ( 3 / 2 ) ∈ ℝ → ( ⌈ ‘ - ( 3 / 2 ) ) = - ( ⌊ ‘ - - ( 3 / 2 ) ) )
6 4 5 ax-mp ⊢ ( ⌈ ‘ - ( 3 / 2 ) ) = - ( ⌊ ‘ - - ( 3 / 2 ) )
7 3 recni ⊢ ( 3 / 2 ) ∈ ℂ
8 7 negnegi ⊢ - - ( 3 / 2 ) = ( 3 / 2 )
9 8 eqcomi ⊢ ( 3 / 2 ) = - - ( 3 / 2 )
10 9 fveq2i ⊢ ( ⌊ ‘ ( 3 / 2 ) ) = ( ⌊ ‘ - - ( 3 / 2 ) )
11 10 eqeq1i ⊢ ( ( ⌊ ‘ ( 3 / 2 ) ) = 1 ↔ ( ⌊ ‘ - - ( 3 / 2 ) ) = 1 )
12 11 biimpi ⊢ ( ( ⌊ ‘ ( 3 / 2 ) ) = 1 → ( ⌊ ‘ - - ( 3 / 2 ) ) = 1 )
13 12 negeqd ⊢ ( ( ⌊ ‘ ( 3 / 2 ) ) = 1 → - ( ⌊ ‘ - - ( 3 / 2 ) ) = - 1 )
14 6 13 eqtrid ⊢ ( ( ⌊ ‘ ( 3 / 2 ) ) = 1 → ( ⌈ ‘ - ( 3 / 2 ) ) = - 1 )
15 ceilval ⊢ ( ( 3 / 2 ) ∈ ℝ → ( ⌈ ‘ ( 3 / 2 ) ) = - ( ⌊ ‘ - ( 3 / 2 ) ) )
16 3 15 ax-mp ⊢ ( ⌈ ‘ ( 3 / 2 ) ) = - ( ⌊ ‘ - ( 3 / 2 ) )
17 negeq ⊢ ( ( ⌊ ‘ - ( 3 / 2 ) ) = - 2 → - ( ⌊ ‘ - ( 3 / 2 ) ) = - - 2 )
18 2cn ⊢ 2 ∈ ℂ
19 18 negnegi ⊢ - - 2 = 2
20 17 19 eqtrdi ⊢ ( ( ⌊ ‘ - ( 3 / 2 ) ) = - 2 → - ( ⌊ ‘ - ( 3 / 2 ) ) = 2 )
21 16 20 eqtrid ⊢ ( ( ⌊ ‘ - ( 3 / 2 ) ) = - 2 → ( ⌈ ‘ ( 3 / 2 ) ) = 2 )
22 14 21 anim12ci ⊢ ( ( ( ⌊ ‘ ( 3 / 2 ) ) = 1 ∧ ( ⌊ ‘ - ( 3 / 2 ) ) = - 2 ) → ( ( ⌈ ‘ ( 3 / 2 ) ) = 2 ∧ ( ⌈ ‘ - ( 3 / 2 ) ) = - 1 ) )
23 1 22 ax-mp ⊢ ( ( ⌈ ‘ ( 3 / 2 ) ) = 2 ∧ ( ⌈ ‘ - ( 3 / 2 ) ) = - 1 )