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Theorem dvdsr 19390
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 19388 . . 3 Rel
32brrelex12i 5584 . 2 (𝑋 𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
4 elex 3487 . . 3 (𝑋𝐵𝑋 ∈ V)
5 id 22 . . . . 5 ((𝑧 · 𝑋) = 𝑌 → (𝑧 · 𝑋) = 𝑌)
6 ovex 7173 . . . . 5 (𝑧 · 𝑋) ∈ V
75, 6eqeltrrdi 2923 . . . 4 ((𝑧 · 𝑋) = 𝑌𝑌 ∈ V)
87rexlimivw 3268 . . 3 (∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌𝑌 ∈ V)
94, 8anim12i 615 . 2 ((𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
10 simpl 486 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
1110eleq1d 2898 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐵𝑋𝐵))
1210oveq2d 7156 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧 · 𝑥) = (𝑧 · 𝑋))
13 simpr 488 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
1412, 13eqeq12d 2838 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧 · 𝑋) = 𝑌))
1514rexbidv 3283 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))
1611, 15anbi12d 633 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌)))
17 dvdsr.1 . . . 4 𝐵 = (Base‘𝑅)
18 dvdsr.3 . . . 4 · = (.r𝑅)
1917, 1, 18dvdsrval 19389 . . 3 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}
2016, 19brabga 5398 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌)))
213, 9, 20pm5.21nii 383 1 (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2114  wrex 3131  Vcvv 3469   class class class wbr 5042  cfv 6334  (class class class)co 7140  Basecbs 16474  .rcmulr 16557  rcdsr 19382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-dvdsr 19385
This theorem is referenced by:  dvdsr2  19391  dvdsrmul  19392  dvdsrcl  19393  dvdsrcl2  19394  dvdsrtr  19396  dvdsrmul1  19397  opprunit  19405  crngunit  19406  subrgdvds  19540  rhmdvdsr  30923
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