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Theorem dvds2lem 10733
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 10723 . . . . . . 7 ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼𝐽 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽))
4 divides 10723 . . . . . . 7 ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐾𝐿 ↔ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿))
53, 4bi2anan9 571 . . . . . 6 (((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → ((𝐼𝐽𝐾𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
61, 2, 5syl2anc 403 . . . . 5 (𝜑 → ((𝐼𝐽𝐾𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
76biimpd 142 . . . 4 (𝜑 → ((𝐼𝐽𝐾𝐿) → (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
8 reeanv 2532 . . . 4 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿))
97, 8syl6ibr 160 . . 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 5622 . . . . . . 7 (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀))
1312eqeq1d 2093 . . . . . 6 (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁))
1413rspcev 2715 . . . . 5 ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)
1510, 11, 14syl6an 1366 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
1615rexlimdvva 2492 . . 3 (𝜑 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
179, 16syld 44 . 2 (𝜑 → ((𝐼𝐽𝐾𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
18 dvds2lem.3 . . 3 (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
19 divides 10723 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
2018, 19syl 14 . 2 (𝜑 → (𝑀𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
2117, 20sylibrd 167 1 (𝜑 → ((𝐼𝐽𝐾𝐿) → 𝑀𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wcel 1436  wrex 2356   class class class wbr 3822  (class class class)co 5615   · cmul 7302  cz 8686  cdvds 10721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-iota 4948  df-fv 4991  df-ov 5618  df-dvds 10722
This theorem is referenced by:  dvds2ln  10754  dvds2add  10755  dvds2sub  10756  dvdstr  10758
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