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Theorem dvds2lem 11528
Description: A lemma to assist theorems of with two antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
dvds2lem.1 (𝜑 → (𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ))
dvds2lem.2 (𝜑 → (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ))
dvds2lem.3 (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
dvds2lem.4 ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑍 ∈ ℤ)
dvds2lem.5 ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → (𝑍 · 𝑀) = 𝑁))
Assertion
Ref Expression
dvds2lem (𝜑 → ((𝐼𝐽𝐾𝐿) → 𝑀𝑁))
Distinct variable groups:   𝑥,𝐼,𝑦   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑍(𝑥,𝑦)

Proof of Theorem dvds2lem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dvds2lem.1 . . . . . 6 (𝜑 → (𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ))
2 dvds2lem.2 . . . . . 6 (𝜑 → (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ))
3 divides 11518 . . . . . . 7 ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼𝐽 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽))
4 divides 11518 . . . . . . 7 ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐾𝐿 ↔ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿))
53, 4bi2anan9 595 . . . . . 6 (((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → ((𝐼𝐽𝐾𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
61, 2, 5syl2anc 408 . . . . 5 (𝜑 → ((𝐼𝐽𝐾𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
76biimpd 143 . . . 4 (𝜑 → ((𝐼𝐽𝐾𝐿) → (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
8 reeanv 2600 . . . 4 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿))
97, 8syl6ibr 161 . . 3 (𝜑 → ((𝐼𝐽𝐾𝐿) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿)))
10 dvds2lem.4 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑍 ∈ ℤ)
11 dvds2lem.5 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → (𝑍 · 𝑀) = 𝑁))
12 oveq1 5784 . . . . . . 7 (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀))
1312eqeq1d 2148 . . . . . 6 (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁))
1413rspcev 2789 . . . . 5 ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)
1510, 11, 14syl6an 1410 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
1615rexlimdvva 2557 . . 3 (𝜑 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
179, 16syld 45 . 2 (𝜑 → ((𝐼𝐽𝐾𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
18 dvds2lem.3 . . 3 (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
19 divides 11518 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
2018, 19syl 14 . 2 (𝜑 → (𝑀𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
2117, 20sylibrd 168 1 (𝜑 → ((𝐼𝐽𝐾𝐿) → 𝑀𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wrex 2417   class class class wbr 3932  (class class class)co 5777   · cmul 7644  cz 9073  cdvds 11516
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-iota 5091  df-fv 5134  df-ov 5780  df-dvds 11517
This theorem is referenced by:  dvds2ln  11549  dvds2add  11550  dvds2sub  11551  dvdstr  11553
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