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Mirrors > Home > ILE Home > Th. List > dvds0lem | GIF version |
Description: A lemma to assist theorems of ∥ with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvds0lem | ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 · 𝑀) = 𝑁) → 𝑀 ∥ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5831 | . . . . . . . . 9 ⊢ (𝑥 = 𝐾 → (𝑥 · 𝑀) = (𝐾 · 𝑀)) | |
2 | 1 | eqeq1d 2166 | . . . . . . . 8 ⊢ (𝑥 = 𝐾 → ((𝑥 · 𝑀) = 𝑁 ↔ (𝐾 · 𝑀) = 𝑁)) |
3 | 2 | rspcev 2816 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ (𝐾 · 𝑀) = 𝑁) → ∃𝑥 ∈ ℤ (𝑥 · 𝑀) = 𝑁) |
4 | 3 | adantl 275 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ (𝐾 · 𝑀) = 𝑁)) → ∃𝑥 ∈ ℤ (𝑥 · 𝑀) = 𝑁) |
5 | divides 11685 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝑀) = 𝑁)) | |
6 | 5 | adantr 274 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ (𝐾 · 𝑀) = 𝑁)) → (𝑀 ∥ 𝑁 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝑀) = 𝑁)) |
7 | 4, 6 | mpbird 166 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ (𝐾 · 𝑀) = 𝑁)) → 𝑀 ∥ 𝑁) |
8 | 7 | expr 373 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐾 ∈ ℤ) → ((𝐾 · 𝑀) = 𝑁 → 𝑀 ∥ 𝑁)) |
9 | 8 | 3impa 1177 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝐾 · 𝑀) = 𝑁 → 𝑀 ∥ 𝑁)) |
10 | 9 | 3comr 1193 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) = 𝑁 → 𝑀 ∥ 𝑁)) |
11 | 10 | imp 123 | 1 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 · 𝑀) = 𝑁) → 𝑀 ∥ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 ∃wrex 2436 class class class wbr 3965 (class class class)co 5824 · cmul 7737 ℤcz 9167 ∥ cdvds 11683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-iota 5135 df-fv 5178 df-ov 5827 df-dvds 11684 |
This theorem is referenced by: iddvds 11699 1dvds 11700 dvds0 11701 dvdsmul1 11708 dvdsmul2 11709 divalgmod 11817 oddpwdclemxy 12043 ex-dvds 13306 |
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