Theorem dvds1lem 15400

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 6929	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀))
4	3	eqeq1d 2779	. . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁))
5	4	rspcev 3510	. . . 4 ⊢ ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)
6	1, 2, 5	syl6an 674	. . 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 15389	. . 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 15389	. . 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 286	1 ⊢ (𝜑 → (𝐽 ∥ 𝐾 → 𝑀 ∥ 𝑁))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2106  ∃wrex 3090   class class class wbr 4886  (class class class)co 6922   · cmul 10277  ℤcz 11728   ∥ cdvds 15387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-iota 6099  df-fv 6143  df-ov 6925  df-dvds 15388

This theorem is referenced by:  negdvdsb  15405  dvdsnegb  15406  muldvds1  15413  muldvds2  15414  dvdscmul  15415  dvdsmulc  15416  dvdscmulr  15417  dvdsmulcr  15418