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Mirrors > Home > MPE Home > Th. List > dvdszrcl | Structured version Visualization version GIF version |
Description: Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
dvdszrcl | ⊢ (𝑋 ∥ 𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dvds 16288 | . . 3 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} | |
2 | opabssxp 5781 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} ⊆ (ℤ × ℤ) | |
3 | 1, 2 | eqsstri 4030 | . 2 ⊢ ∥ ⊆ (ℤ × ℤ) |
4 | 3 | brel 5754 | 1 ⊢ (𝑋 ∥ 𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 class class class wbr 5148 {copab 5210 × cxp 5687 (class class class)co 7431 · cmul 11158 ℤcz 12611 ∥ cdvds 16287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-dvds 16288 |
This theorem is referenced by: dvdsmod0 16293 p1modz1 16294 dvdsmodexp 16295 dvdsaddre2b 16341 dvdsabseq 16347 divconjdvds 16349 evenelz 16370 4dvdseven 16407 dfgcd2 16580 dvdsmulgcd 16590 dvdsnprmd 16724 dvdszzq 16755 oddvdsi 19581 odmulg 19589 gexdvdsi 19616 dvdschrmulg 21561 nnproddivdvdsd 41982 lcmineqlem14 42024 aks6d1c6isolem3 42158 grpods 42176 nzss 44313 nzin 44314 |
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