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Mathbox for Jeff Hoffman |
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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 3817 | . . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ 𝐷) |
3 | 1 | sseli 3817 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ 𝐷) |
4 | nnssi2.2 | . . . 4 ⊢ (𝐵 ∈ ℕ → 𝜑) | |
5 | 2, 3, 4 | 3anim123i 1151 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑)) |
6 | 5 | 3anidm23 1493 | . 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 386 ∧ w3a 1071 ∈ wcel 2107 ⊆ wss 3792 ℕcn 11379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-in 3799 df-ss 3806 |
This theorem is referenced by: nndivsub 33047 |
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