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Metamath Proof Explorer

Theorem dpgti

Description: Comparing a decimal expansions with the next lower integer. (Contributed by Thierry Arnoux, 16-Dec-2021)

Ref Expression
Hypotheses dpgti.a ⊢ 𝐴 ∈ ℕ0
dpgti.b ⊢ 𝐵 ∈ ℝ+
Assertion dpgti 𝐴 < ( 𝐴 . 𝐵 )

Proof

Step Hyp Ref Expression
1 dpgti.a ⊢ 𝐴 ∈ ℕ0
2 dpgti.b ⊢ 𝐵 ∈ ℝ+
3 1 nn0rei ⊢ 𝐴 ∈ ℝ
4 10re ⊢ 1 0 ∈ ℝ
5 10pos ⊢ 0 < 1 0
6 4 5 pm3.2i ⊢ ( 1 0 ∈ ℝ ∧ 0 < 1 0 )
7 elrp ⊢ ( 1 0 ∈ ℝ+ ↔ ( 1 0 ∈ ℝ ∧ 0 < 1 0 ) )
8 6 7 mpbir ⊢ 1 0 ∈ ℝ+
9 rpdivcl ⊢ ( ( 𝐵 ∈ ℝ+ ∧ 1 0 ∈ ℝ+ ) → ( 𝐵 / 1 0 ) ∈ ℝ+ )
10 2 8 9 mp2an ⊢ ( 𝐵 / 1 0 ) ∈ ℝ+
11 ltaddrp ⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 / 1 0 ) ∈ ℝ+ ) → 𝐴 < ( 𝐴 + ( 𝐵 / 1 0 ) ) )
12 3 10 11 mp2an ⊢ 𝐴 < ( 𝐴 + ( 𝐵 / 1 0 ) )
13 rpre ⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ )
14 2 13 ax-mp ⊢ 𝐵 ∈ ℝ
15 1 14 dpval2 ⊢ ( 𝐴 . 𝐵 ) = ( 𝐴 + ( 𝐵 / 1 0 ) )
16 12 15 breqtrri ⊢ 𝐴 < ( 𝐴 . 𝐵 )