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Theorem dvdsr 18846
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 18844 . . 3 Rel
3 brrelex12 5312 . . 3 ((Rel 𝑋 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
42, 3mpan 708 . 2 (𝑋 𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
5 elex 3352 . . 3 (𝑋𝐵𝑋 ∈ V)
6 id 22 . . . . 5 ((𝑧 · 𝑋) = 𝑌 → (𝑧 · 𝑋) = 𝑌)
7 ovex 6841 . . . . 5 (𝑧 · 𝑋) ∈ V
86, 7syl6eqelr 2848 . . . 4 ((𝑧 · 𝑋) = 𝑌𝑌 ∈ V)
98rexlimivw 3167 . . 3 (∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌𝑌 ∈ V)
105, 9anim12i 591 . 2 ((𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
11 simpl 474 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
1211eleq1d 2824 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐵𝑋𝐵))
1311oveq2d 6829 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧 · 𝑥) = (𝑧 · 𝑋))
14 simpr 479 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
1513, 14eqeq12d 2775 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧 · 𝑋) = 𝑌))
1615rexbidv 3190 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))
1712, 16anbi12d 749 . . 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 18845 . . 3 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}
2117, 20brabga 5139 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌)))
224, 10, 21pm5.21nii 367 1 (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  wrex 3051  Vcvv 3340   class class class wbr 4804  Rel wrel 5271  cfv 6049  (class class class)co 6813  Basecbs 16059  .rcmulr 16144  rcdsr 18838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-dvdsr 18841
This theorem is referenced by:  dvdsr2  18847  dvdsrmul  18848  dvdsrcl  18849  dvdsrcl2  18850  dvdsrtr  18852  dvdsrmul1  18853  opprunit  18861  crngunit  18862  subrgdvds  18996  rhmdvdsr  30127
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