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Theorem dvds1lem 15202
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 6800 . . . . . 6 (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀))
43eqeq1d 2773 . . . . 5 (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁))
54rspcev 3460 . . . 4 ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)
61, 2, 5syl6an 663 . . 3 ((𝜑𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
76rexlimdva 3179 . 2 (𝜑 → (∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
8 dvds1lem.1 . . 3 (𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ))
9 divides 15191 . . 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 15191 . . 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 382   = wceq 1631  wcel 2145  wrex 3062   class class class wbr 4786  (class class class)co 6793   · cmul 10143  cz 11579  cdvds 15189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-iota 5994  df-fv 6039  df-ov 6796  df-dvds 15190
This theorem is referenced by:  negdvdsb  15207  dvdsnegb  15208  muldvds1  15215  muldvds2  15216  dvdscmul  15217  dvdsmulc  15218  dvdscmulr  15219  dvdsmulcr  15220
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