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Theorem dvds2lem 15485
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 15472 . . . . . . 7 ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼𝐽 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽))
4 divides 15472 . . . . . . 7 ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐾𝐿 ↔ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿))
53, 4bi2anan9 626 . . . . . 6 (((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → ((𝐼𝐽𝐾𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
61, 2, 5syl2anc 576 . . . . 5 (𝜑 → ((𝐼𝐽𝐾𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
76biimpd 221 . . . 4 (𝜑 → ((𝐼𝐽𝐾𝐿) → (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
8 reeanv 3308 . . . 4 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿))
97, 8syl6ibr 244 . . 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 6985 . . . . . . 7 (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀))
1312eqeq1d 2780 . . . . . 6 (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁))
1413rspcev 3535 . . . . 5 ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)
1510, 11, 14syl6an 671 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
1615rexlimdvva 3239 . . 3 (𝜑 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
179, 16syld 47 . 2 (𝜑 → ((𝐼𝐽𝐾𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
18 dvds2lem.3 . . 3 (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
19 divides 15472 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
2018, 19syl 17 . 2 (𝜑 → (𝑀𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
2117, 20sylibrd 251 1 (𝜑 → ((𝐼𝐽𝐾𝐿) → 𝑀𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wrex 3089   class class class wbr 4930  (class class class)co 6978   · cmul 10342  cz 11796  cdvds 15470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-iota 6154  df-fv 6198  df-ov 6981  df-dvds 15471
This theorem is referenced by:  dvds2ln  15505  dvds2add  15506  dvds2sub  15507  dvdstr  15509
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