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Theorem dvds1lem 11790
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 5876 . . . . . 6 (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀))
43eqeq1d 2186 . . . . 5 (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁))
54rspcev 2841 . . . 4 ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)
61, 2, 5syl6an 1434 . . 3 ((𝜑𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
76rexlimdva 2594 . 2 (𝜑 → (∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
8 dvds1lem.1 . . 3 (𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ))
9 divides 11777 . . 3 ((𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐽𝐾 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾))
108, 9syl 14 . 2 (𝜑 → (𝐽𝐾 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾))
11 dvds1lem.2 . . 3 (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
12 divides 11777 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
1311, 12syl 14 . 2 (𝜑 → (𝑀𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
147, 10, 133imtr4d 203 1 (𝜑 → (𝐽𝐾𝑀𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wrex 2456   class class class wbr 4000  (class class class)co 5869   · cmul 7804  cz 9239  cdvds 11775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-iota 5174  df-fv 5220  df-ov 5872  df-dvds 11776
This theorem is referenced by:  negdvdsb  11795  dvdsnegb  11796  muldvds1  11804  muldvds2  11805  dvdscmul  11806  dvdsmulc  11807  dvdscmulr  11808  dvdsmulcr  11809
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