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Theorem dvdsr 19033
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 19031 . . 3 Rel
32brrelex12i 5405 . 2 (𝑋 𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
4 elex 3414 . . 3 (𝑋𝐵𝑋 ∈ V)
5 id 22 . . . . 5 ((𝑧 · 𝑋) = 𝑌 → (𝑧 · 𝑋) = 𝑌)
6 ovex 6954 . . . . 5 (𝑧 · 𝑋) ∈ V
75, 6syl6eqelr 2868 . . . 4 ((𝑧 · 𝑋) = 𝑌𝑌 ∈ V)
87rexlimivw 3211 . . 3 (∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌𝑌 ∈ V)
94, 8anim12i 606 . 2 ((𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
10 simpl 476 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
1110eleq1d 2844 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐵𝑋𝐵))
1210oveq2d 6938 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧 · 𝑥) = (𝑧 · 𝑋))
13 simpr 479 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
1412, 13eqeq12d 2793 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧 · 𝑋) = 𝑌))
1514rexbidv 3237 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))
1611, 15anbi12d 624 . . 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 19032 . . 3 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}
2016, 19brabga 5226 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌)))
213, 9, 20pm5.21nii 370 1 (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1601  wcel 2107  wrex 3091  Vcvv 3398   class class class wbr 4886  cfv 6135  (class class class)co 6922  Basecbs 16255  .rcmulr 16339  rcdsr 19025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-dvdsr 19028
This theorem is referenced by:  dvdsr2  19034  dvdsrmul  19035  dvdsrcl  19036  dvdsrcl2  19037  dvdsrtr  19039  dvdsrmul1  19040  opprunit  19048  crngunit  19049  subrgdvds  19186  rhmdvdsr  30380
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