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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 16291 | . . 3 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} | |
| 2 | opabssxp 5778 | . . 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 5750 | 1 ⊢ (𝑋 ∥ 𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 {copab 5205 × cxp 5683 (class class class)co 7431 · cmul 11160 ℤcz 12613 ∥ cdvds 16290 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-dvds 16291 |
| This theorem is referenced by: dvdsmod0 16296 p1modz1 16297 dvdsmodexp 16298 dvdsaddre2b 16344 dvdsabseq 16350 divconjdvds 16352 evenelz 16373 4dvdseven 16410 dfgcd2 16583 dvdsmulgcd 16593 dvdsnprmd 16727 dvdszzq 16758 oddvdsi 19566 odmulg 19574 gexdvdsi 19601 dvdschrmulg 21543 nnproddivdvdsd 42001 lcmineqlem14 42043 aks6d1c6isolem3 42177 grpods 42195 nzss 44336 nzin 44337 |
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