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Theorem dvds1lem 10119
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 5547 . . . . . 6 (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀))
43eqeq1d 2064 . . . . 5 (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁))
54rspcev 2673 . . . 4 ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)
61, 2, 5syl6an 1339 . . 3 ((𝜑𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
76rexlimdva 2450 . 2 (𝜑 → (∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁))
8 dvds1lem.1 . . 3 (𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ))
9 divides 10110 . . 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 10110 . . 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 196 1 (𝜑 → (𝐽𝐾𝑀𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wrex 2324   class class class wbr 3792  (class class class)co 5540   · cmul 6952  cz 8302  cdvds 10108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-iota 4895  df-fv 4938  df-ov 5543  df-dvds 10109
This theorem is referenced by:  negdvdsb  10124  dvdsnegb  10125  muldvds1  10132  muldvds2  10133  dvdscmul  10134  dvdsmulc  10135  dvdscmulr  10136  dvdsmulcr  10137
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