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Metamath Proof Explorer

Theorem numlt

Description: Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014)

Ref Expression
Hypotheses numlt.1 ⊢ 𝑇 ∈ ℕ
numlt.2 ⊢ 𝐴 ∈ ℕ0
numlt.3 ⊢ 𝐵 ∈ ℕ0
numlt.4 ⊢ 𝐶 ∈ ℕ
numlt.5 ⊢ 𝐵 < 𝐶
Assertion numlt ( ( 𝑇 · 𝐴 ) + 𝐵 ) < ( ( 𝑇 · 𝐴 ) + 𝐶 )

Proof

Step Hyp Ref Expression
1 numlt.1 ⊢ 𝑇 ∈ ℕ
2 numlt.2 ⊢ 𝐴 ∈ ℕ0
3 numlt.3 ⊢ 𝐵 ∈ ℕ0
4 numlt.4 ⊢ 𝐶 ∈ ℕ
5 numlt.5 ⊢ 𝐵 < 𝐶
6 3 nn0rei ⊢ 𝐵 ∈ ℝ
7 4 nnrei ⊢ 𝐶 ∈ ℝ
8 1 nnnn0i ⊢ 𝑇 ∈ ℕ0
9 8 2 nn0mulcli ⊢ ( 𝑇 · 𝐴 ) ∈ ℕ0
10 9 nn0rei ⊢ ( 𝑇 · 𝐴 ) ∈ ℝ
11 6 7 10 ltadd2i ⊢ ( 𝐵 < 𝐶 ↔ ( ( 𝑇 · 𝐴 ) + 𝐵 ) < ( ( 𝑇 · 𝐴 ) + 𝐶 ) )
12 5 11 mpbi ⊢ ( ( 𝑇 · 𝐴 ) + 𝐵 ) < ( ( 𝑇 · 𝐴 ) + 𝐶 )