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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 6556 | . . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀)) | |
4 | 3 | eqeq1d 2612 | . . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁)) |
5 | 4 | rspcev 3282 | . . . 4 ⊢ ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁) |
6 | 1, 2, 5 | syl6an 566 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)) |
7 | 6 | rexlimdva 3013 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)) |
8 | dvds1lem.1 | . . 3 ⊢ (𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) | |
9 | divides 14823 | . . 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 14823 | . . 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 282 | 1 ⊢ (𝜑 → (𝐽 ∥ 𝐾 → 𝑀 ∥ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 (class class class)co 6549 · cmul 9820 ℤcz 11254 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-iota 5768 df-fv 5812 df-ov 6552 df-dvds 14822 |
This theorem is referenced by: negdvdsb 14836 dvdsnegb 14837 muldvds1 14844 muldvds2 14845 dvdscmul 14846 dvdsmulc 14847 dvdscmulr 14848 dvdsmulcr 14849 |
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