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Theorem dvdsr 20435
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 20433 . . 3 Rel
32brrelex12i 5707 . 2 (𝑋 𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
4 elex 3478 . . 3 (𝑋𝐵𝑋 ∈ V)
5 id 23 . . . . 5 ((𝑧 · 𝑋) = 𝑌 → (𝑧 · 𝑋) = 𝑌)
6 ovex 7433 . . . . 5 (𝑧 · 𝑋) ∈ V
75, 6eqeltrrdi 2874 . . . 4 ((𝑧 · 𝑋) = 𝑌𝑌 ∈ V)
87rexlimivw 3162 . . 3 (∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌𝑌 ∈ V)
94, 8anim12i 624 . 2 ((𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
10 simpl 487 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
1110eleq1d 2850 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐵𝑋𝐵))
1210oveq2d 7416 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧 · 𝑥) = (𝑧 · 𝑋))
13 simpr 489 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
1412, 13eqeq12d 2781 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧 · 𝑋) = 𝑌))
1514rexbidv 3189 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))
1611, 15anbi12d 643 . . 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 20434 . . 3 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}
2016, 19brabga 5509 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌)))
213, 9, 20pm5.21nii 381 1 (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 400   = wceq 1563  wcel 2145  wrex 3089  Vcvv 3457   class class class wbr 5105  cfv 6525  (class class class)co 7400  Basecbs 17259  .rcmulr 17301  rcdsr 20427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-dvdsr 20430
This theorem is referenced by:  dvdsr2  20436  dvdsrmul  20437  dvdsrcl  20438  dvdsrcl2  20439  dvdsrtr  20441  dvdsrmul1  20442  opprunit  20450  crngunit  20451  rhmdvdsr  20582  subrgdvds  20662  isunit2  33472  dvdsruassoi  33613  dvdsruasso  33614  dvdsrspss  33616  rprmasso2  33733  unitmulrprm  33735  rprmirredlem  33737  1arithufdlem3  33753  rhmqusspan  42814  unitscyglem5  42828
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