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Theorem dvdsr 18411
Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
dvdsr.1 𝐵 = (Base‘𝑅)
dvdsr.2 = (∥r𝑅)
dvdsr.3 · = (.r𝑅)
Assertion
Ref Expression
dvdsr (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))
Distinct variable groups:   𝑧,𝐵   𝑧,𝑋   𝑧,𝑌   𝑧,𝑅   𝑧, ·
Allowed substitution hint:   (𝑧)

Proof of Theorem dvdsr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvdsr.2 . . . 4 = (∥r𝑅)
21reldvdsr 18409 . . 3 Rel
3 brrelex12 5065 . . 3 ((Rel 𝑋 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
42, 3mpan 701 . 2 (𝑋 𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
5 elex 3180 . . 3 (𝑋𝐵𝑋 ∈ V)
6 id 22 . . . . 5 ((𝑧 · 𝑋) = 𝑌 → (𝑧 · 𝑋) = 𝑌)
7 ovex 6551 . . . . 5 (𝑧 · 𝑋) ∈ V
86, 7syl6eqelr 2692 . . . 4 ((𝑧 · 𝑋) = 𝑌𝑌 ∈ V)
98rexlimivw 3006 . . 3 (∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌𝑌 ∈ V)
105, 9anim12i 587 . 2 ((𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
11 simpl 471 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
1211eleq1d 2667 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐵𝑋𝐵))
1311oveq2d 6539 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧 · 𝑥) = (𝑧 · 𝑋))
14 simpr 475 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
1513, 14eqeq12d 2620 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧 · 𝑋) = 𝑌))
1615rexbidv 3029 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))
1712, 16anbi12d 742 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌)))
18 dvdsr.1 . . . 4 𝐵 = (Base‘𝑅)
19 dvdsr.3 . . . 4 · = (.r𝑅)
2018, 1, 19dvdsrval 18410 . . 3 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}
2117, 20brabga 4900 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌)))
224, 10, 21pm5.21nii 366 1 (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1975  wrex 2892  Vcvv 3168   class class class wbr 4573  Rel wrel 5029  cfv 5786  (class class class)co 6523  Basecbs 15637  .rcmulr 15711  rcdsr 18403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-dvdsr 18406
This theorem is referenced by:  dvdsr2  18412  dvdsrmul  18413  dvdsrcl  18414  dvdsrcl2  18415  dvdsrtr  18417  dvdsrmul1  18418  opprunit  18426  crngunit  18427  subrgdvds  18559  rhmdvdsr  28951
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