Theorem dvds1lem 15905

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	⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑍 ∈ ℤ)
dvds1lem.4	⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → (𝑍 · 𝑀) = 𝑁))

Assertion

Ref	Expression
dvds1lem	⊢ (𝜑 → (𝐽 ∥ 𝐾 → 𝑀 ∥ 𝑁))

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.

Step	Hyp	Ref	Expression
1		dvds1lem.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑍 ∈ ℤ)
2		dvds1lem.4	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → (𝑍 · 𝑀) = 𝑁))
3		oveq1 7262	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀))
4	3	eqeq1d 2740	. . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁))
5	4	rspcev 3552	. . . 4 ⊢ ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)
6	1, 2, 5	syl6an 680	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
7	6	rexlimdva 3212	. 2 ⊢ (𝜑 → (∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
8		dvds1lem.1	. . 3 ⊢ (𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ))
9		divides 15893	. . 3 ⊢ ((𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐽 ∥ 𝐾 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾))
10	8, 9	syl 17	. 2 ⊢ (𝜑 → (𝐽 ∥ 𝐾 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾))
11		dvds1lem.2	. . 3 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
12		divides 15893	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
13	11, 12	syl 17	. 2 ⊢ (𝜑 → (𝑀 ∥ 𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
14	7, 10, 13	3imtr4d 293	1 ⊢ (𝜑 → (𝐽 ∥ 𝐾 → 𝑀 ∥ 𝑁))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   class class class wbr 5070  (class class class)co 7255   · cmul 10807  ℤcz 12249   ∥ cdvds 15891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-iota 6376  df-fv 6426  df-ov 7258  df-dvds 15892

This theorem is referenced by:  negdvdsb  15910  dvdsnegb  15911  muldvds1  15918  muldvds2  15919  dvdscmul  15920  dvdsmulc  15921  dvdscmulr  15922  dvdsmulcr  15923