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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 15603 | . . 3 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} | |
2 | opabssxp 5636 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} ⊆ (ℤ × ℤ) | |
3 | 1, 2 | eqsstri 3994 | . 2 ⊢ ∥ ⊆ (ℤ × ℤ) |
4 | 3 | brel 5610 | 1 ⊢ (𝑋 ∥ 𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 class class class wbr 5059 {copab 5121 × cxp 5546 (class class class)co 7149 · cmul 10535 ℤcz 11975 ∥ cdvds 15602 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-xp 5554 df-dvds 15603 |
This theorem is referenced by: dvdsmod0 15608 p1modz1 15609 dvdsmodexp 15610 dvdsaddre2b 15652 dvdsabseq 15658 divconjdvds 15660 evenelz 15680 4dvdseven 15719 dfgcd2 15889 dvdsmulgcd 15900 dvdsnprmd 16029 oddvdsi 18671 odmulg 18678 gexdvdsi 18703 dvdszzq 30531 dvdschrmulg 30879 nzss 40723 nzin 40724 |
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