ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dvds2lem GIF version

Theorem dvds2lem 11781
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 11767 . . . . . . 7 ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼𝐽 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽))
4 divides 11767 . . . . . . 7 ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐾𝐿 ↔ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿))
53, 4bi2anan9 606 . . . . . 6 (((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → ((𝐼𝐽𝐾𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
61, 2, 5syl2anc 411 . . . . 5 (𝜑 → ((𝐼𝐽𝐾𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
76biimpd 144 . . . 4 (𝜑 → ((𝐼𝐽𝐾𝐿) → (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
8 reeanv 2646 . . . 4 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿))
97, 8syl6ibr 162 . . 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 5875 . . . . . . 7 (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀))
1312eqeq1d 2186 . . . . . 6 (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁))
1413rspcev 2841 . . . . 5 ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)
1510, 11, 14syl6an 1434 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
1615rexlimdvva 2602 . . 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 11767 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
2018, 19syl 14 . 2 (𝜑 → (𝑀𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
2117, 20sylibrd 169 1 (𝜑 → ((𝐼𝐽𝐾𝐿) → 𝑀𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wrex 2456   class class class wbr 4000  (class class class)co 5868   · cmul 7794  cz 9229  cdvds 11765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-iota 5173  df-fv 5219  df-ov 5871  df-dvds 11766
This theorem is referenced by:  dvds2ln  11802  dvds2add  11803  dvds2sub  11804  dvdstr  11806
  Copyright terms: Public domain W3C validator