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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnssi2 | Structured version Visualization version GIF version |
Description: Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
Ref | Expression |
---|---|
nnssi2.1 | ⊢ ℕ ⊆ 𝐷 |
nnssi2.2 | ⊢ (𝐵 ∈ ℕ → 𝜑) |
nnssi2.3 | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑) → 𝜓) |
Ref | Expression |
---|---|
nnssi2 | ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssi2.1 | . . . . 5 ⊢ ℕ ⊆ 𝐷 | |
2 | 1 | sseli 3922 | . . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ 𝐷) |
3 | 1 | sseli 3922 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ 𝐷) |
4 | nnssi2.2 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝜑) | |
5 | 2, 3, 4 | 3anim123i 1150 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑)) |
6 | 5 | 3anidm23 1420 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑)) |
7 | nnssi2.3 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑) → 𝜓) | |
8 | 6, 7 | syl 17 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2110 ⊆ wss 3892 ℕcn 11984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1088 df-tru 1545 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-v 3433 df-in 3899 df-ss 3909 |
This theorem is referenced by: nndivsub 34655 |
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