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Theorem dvds1lem 16225
Description: A lemma to assist theorems of with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
dvds1lem.1 (𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ))
dvds1lem.2 (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
dvds1lem.3 ((𝜑𝑥 ∈ ℤ) → 𝑍 ∈ ℤ)
dvds1lem.4 ((𝜑𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → (𝑍 · 𝑀) = 𝑁))
Assertion
Ref Expression
dvds1lem (𝜑 → (𝐽𝐾𝑀𝑁))
Distinct variable groups:   𝑥,𝐽   𝑥,𝐾   𝑥,𝑀   𝑥,𝑁   𝜑,𝑥
Allowed substitution hint:   𝑍(𝑥)

Proof of Theorem dvds1lem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dvds1lem.3 . . . 4 ((𝜑𝑥 ∈ ℤ) → 𝑍 ∈ ℤ)
2 dvds1lem.4 . . . 4 ((𝜑𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → (𝑍 · 𝑀) = 𝑁))
3 oveq1 7363 . . . . . 6 (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀))
43eqeq1d 2737 . . . . 5 (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁))
54rspcev 3562 . . . 4 ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)
61, 2, 5syl6an 685 . . 3 ((𝜑𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
76rexlimdva 3136 . 2 (𝜑 → (∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
8 dvds1lem.1 . . 3 (𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ))
9 divides 16212 . . 3 ((𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐽𝐾 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾))
108, 9syl 17 . 2 (𝜑 → (𝐽𝐾 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾))
11 dvds1lem.2 . . 3 (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
12 divides 16212 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
1311, 12syl 17 . 2 (𝜑 → (𝑀𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
147, 10, 133imtr4d 294 1 (𝜑 → (𝐽𝐾𝑀𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1542  wcel 2114  wrex 3059   class class class wbr 5074  (class class class)co 7356   · cmul 11032  cz 12513  cdvds 16210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-pr 5364
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-iota 6443  df-fv 6495  df-ov 7359  df-dvds 16211
This theorem is referenced by:  negdvdsb  16230  dvdsnegb  16231  muldvds1  16238  muldvds2  16239  dvdscmul  16240  dvdsmulc  16241  dvdscmulr  16242  dvdsmulcr  16243
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