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Theorem dvds1lem 15040
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 6697 . . . . . 6 (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀))
43eqeq1d 2653 . . . . 5 (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁))
54rspcev 3340 . . . 4 ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)
61, 2, 5syl6an 567 . . 3 ((𝜑𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
76rexlimdva 3060 . 2 (𝜑 → (∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
8 dvds1lem.1 . . 3 (𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ))
9 divides 15029 . . 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 15029 . . 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 283 1 (𝜑 → (𝐽𝐾𝑀𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942   class class class wbr 4685  (class class class)co 6690   · cmul 9979  cz 11415  cdvds 15027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-iota 5889  df-fv 5934  df-ov 6693  df-dvds 15028
This theorem is referenced by:  negdvdsb  15045  dvdsnegb  15046  muldvds1  15053  muldvds2  15054  dvdscmul  15055  dvdsmulc  15056  dvdscmulr  15057  dvdsmulcr  15058
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