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Mirrors > Home > MPE Home > Th. List > dvds1lem | Structured version Visualization version GIF version |
Description: A lemma to assist theorems of ∥ with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvds1lem.1 | ⊢ (𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) |
dvds1lem.2 | ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
dvds1lem.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑍 ∈ ℤ) |
dvds1lem.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → (𝑍 · 𝑀) = 𝑁)) |
Ref | Expression |
---|---|
dvds1lem | ⊢ (𝜑 → (𝐽 ∥ 𝐾 → 𝑀 ∥ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvds1lem.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → 𝑍 ∈ ℤ) | |
2 | dvds1lem.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → (𝑍 · 𝑀) = 𝑁)) | |
3 | oveq1 7419 | . . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀)) | |
4 | 3 | eqeq1d 2733 | . . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁)) |
5 | 4 | rspcev 3612 | . . . 4 ⊢ ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁) |
6 | 1, 2, 5 | syl6an 681 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)) |
7 | 6 | rexlimdva 3154 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)) |
8 | dvds1lem.1 | . . 3 ⊢ (𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) | |
9 | divides 16206 | . . 3 ⊢ ((𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐽 ∥ 𝐾 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾)) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → (𝐽 ∥ 𝐾 ↔ ∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾)) |
11 | dvds1lem.2 | . . 3 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
12 | divides 16206 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝑀 ∥ 𝑁 ↔ ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)) |
14 | 7, 10, 13 | 3imtr4d 294 | 1 ⊢ (𝜑 → (𝐽 ∥ 𝐾 → 𝑀 ∥ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 class class class wbr 5148 (class class class)co 7412 · cmul 11121 ℤcz 12565 ∥ cdvds 16204 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-iota 6495 df-fv 6551 df-ov 7415 df-dvds 16205 |
This theorem is referenced by: negdvdsb 16223 dvdsnegb 16224 muldvds1 16231 muldvds2 16232 dvdscmul 16233 dvdsmulc 16234 dvdscmulr 16235 dvdsmulcr 16236 |
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