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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 6657 | . . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 · 𝑀) = (𝑍 · 𝑀)) | |
4 | 3 | eqeq1d 2624 | . . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑧 · 𝑀) = 𝑁 ↔ (𝑍 · 𝑀) = 𝑁)) |
5 | 4 | rspcev 3309 | . . . 4 ⊢ ((𝑍 ∈ ℤ ∧ (𝑍 · 𝑀) = 𝑁) → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁) |
6 | 1, 2, 5 | syl6an 568 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)) |
7 | 6 | rexlimdva 3031 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ ℤ (𝑥 · 𝐽) = 𝐾 → ∃𝑧 ∈ ℤ (𝑧 · 𝑀) = 𝑁)) |
8 | dvds1lem.1 | . . 3 ⊢ (𝜑 → (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) | |
9 | divides 14985 | . . 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 14985 | . . 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 283 | 1 ⊢ (𝜑 → (𝐽 ∥ 𝐾 → 𝑀 ∥ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 class class class wbr 4653 (class class class)co 6650 · cmul 9941 ℤcz 11377 ∥ cdvds 14983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-iota 5851 df-fv 5896 df-ov 6653 df-dvds 14984 |
This theorem is referenced by: negdvdsb 14998 dvdsnegb 14999 muldvds1 15006 muldvds2 15007 dvdscmul 15008 dvdsmulc 15009 dvdscmulr 15010 dvdsmulcr 15011 |
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