Theorem dvds2lem 15624

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 → ((𝐼 ∥ 𝐽 ∧ 𝐾 ∥ 𝐿) → 𝑀 ∥ 𝑁))

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.

Step	Hyp	Ref	Expression
1		dvds2lem.1	. . . . . 6 ⊢ (𝜑 → (𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ))
2		dvds2lem.2	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ))
3		divides 15611	. . . . . . 7 ⊢ ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼 ∥ 𝐽 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽))
4		divides 15611	. . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐾 ∥ 𝐿 ↔ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿))
5	3, 4	bi2anan9 637	. . . . . 6 ⊢ (((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ)) → ((𝐼 ∥ 𝐽 ∧ 𝐾 ∥ 𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
6	1, 2, 5	syl2anc 586	. . . . 5 ⊢ (𝜑 → ((𝐼 ∥ 𝐽 ∧ 𝐾 ∥ 𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
7	6	biimpd 231	. . . 4 ⊢ (𝜑 → ((𝐼 ∥ 𝐽 ∧ 𝐾 ∥ 𝐿) → (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿)))
8		reeanv 3369	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) ↔ (∃𝑥 ∈ ℤ (𝑥 · 𝐼) = 𝐽 ∧ ∃𝑦 ∈ ℤ (𝑦 · 𝐾) = 𝐿))
9	7, 8	syl6ibr 254	. . 3 ⊢ (𝜑 → ((𝐼 ∥ 𝐽 ∧ 𝐾 ∥ 𝐿) → ∃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 7165	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀))
13	12	eqeq1d 2825	. . . . . 6 ⊢ (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁))
14	13	rspcev 3625	. . . . 5 ⊢ ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)
15	10, 11, 14	syl6an 682	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
16	15	rexlimdvva 3296	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 · 𝐼) = 𝐽 ∧ (𝑦 · 𝐾) = 𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
17	9, 16	syld 47	. 2 ⊢ (𝜑 → ((𝐼 ∥ 𝐽 ∧ 𝐾 ∥ 𝐿) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
18		dvds2lem.3	. . 3 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
19		divides 15611	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
20	18, 19	syl 17	. 2 ⊢ (𝜑 → (𝑀 ∥ 𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
21	17, 20	sylibrd 261	1 ⊢ (𝜑 → ((𝐼 ∥ 𝐽 ∧ 𝐾 ∥ 𝐿) → 𝑀 ∥ 𝑁))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3141   class class class wbr 5068  (class class class)co 7158   · cmul 10544  ℤcz 11984   ∥ cdvds 15609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-iota 6316  df-fv 6365  df-ov 7161  df-dvds 15610

This theorem is referenced by:  dvds2ln  15644  dvds2add  15645  dvds2sub  15646  dvdstr  15648