Mathbox for Jeff Hoffman |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > nnssi3 | 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 |
---|---|
nnssi3.1 | ⊢ ℕ ⊆ 𝐷 |
nnssi3.2 | ⊢ (𝐶 ∈ ℕ → 𝜑) |
nnssi3.3 | ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ∧ 𝜑) → 𝜓) |
Ref | Expression |
---|---|
nnssi3 | ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssi3.1 | . . . 4 ⊢ ℕ ⊆ 𝐷 | |
2 | 1 | sseli 3928 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ 𝐷) |
3 | 1 | sseli 3928 | . . 3 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ 𝐷) |
4 | 1 | sseli 3928 | . . 3 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ 𝐷) |
5 | 2, 3, 4 | 3anim123i 1150 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷)) |
6 | nnssi3.2 | . . 3 ⊢ (𝐶 ∈ ℕ → 𝜑) | |
7 | 6 | 3ad2ant3 1134 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜑) |
8 | nnssi3.3 | . 2 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ∧ 𝜑) → 𝜓) | |
9 | 5, 7, 8 | syl2anc 584 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2105 ⊆ wss 3898 ℕcn 12074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1088 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3443 df-in 3905 df-ss 3915 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |