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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 3564 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ 𝐷) |
3 | 1 | sseli 3564 | . . 3 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ 𝐷) |
4 | 1 | sseli 3564 | . . 3 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ 𝐷) |
5 | 2, 3, 4 | 3anim123i 1240 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷)) |
6 | nnssi3.2 | . . 3 ⊢ (𝐶 ∈ ℕ → 𝜑) | |
7 | 6 | 3ad2ant3 1077 | . 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 691 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ⊆ wss 3540 ℕcn 10897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 |
This theorem is referenced by: (None) |
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