Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvdsrval Structured version   Visualization version   GIF version

Theorem dvdsrval 19409
 Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
dvdsr.1 𝐵 = (Base‘𝑅)
dvdsr.2 = (∥r𝑅)
dvdsr.3 · = (.r𝑅)
Assertion
Ref Expression
dvdsrval = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}
Distinct variable groups:   𝑥,𝑦,   𝑥,𝑧,𝐵,𝑦   𝑥,𝑅,𝑦,𝑧   𝑥, · ,𝑦,𝑧
Allowed substitution hint:   (𝑧)

Proof of Theorem dvdsrval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dvdsr.2 . . 3 = (∥r𝑅)
2 fveq2 6652 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 dvdsr.1 . . . . . . . . 9 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2851 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
54eleq2d 2875 . . . . . . 7 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥𝐵))
64rexeqdv 3365 . . . . . . 7 (𝑟 = 𝑅 → (∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦))
75, 6anbi12d 633 . . . . . 6 (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦)))
8 fveq2 6652 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
9 dvdsr.3 . . . . . . . . . . 11 · = (.r𝑅)
108, 9eqtr4di 2851 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = · )
1110oveqd 7159 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑧(.r𝑟)𝑥) = (𝑧 · 𝑥))
1211eqeq1d 2800 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑧(.r𝑟)𝑥) = 𝑦 ↔ (𝑧 · 𝑥) = 𝑦))
1312rexbidv 3256 . . . . . . 7 (𝑟 = 𝑅 → (∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦 ↔ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦))
1413anbi2d 631 . . . . . 6 (𝑟 = 𝑅 → ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)))
157, 14bitrd 282 . . . . 5 (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)))
1615opabbidv 5099 . . . 4 (𝑟 = 𝑅 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
17 df-dvdsr 19405 . . . 4 r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦)})
183fvexi 6666 . . . . 5 𝐵 ∈ V
19 eqcom 2805 . . . . . . . . 9 ((𝑧 · 𝑥) = 𝑦𝑦 = (𝑧 · 𝑥))
2019rexbii 3210 . . . . . . . 8 (∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥))
2120abbii 2863 . . . . . . 7 {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} = {𝑦 ∣ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥)}
2218abrexex 7655 . . . . . . 7 {𝑦 ∣ ∃𝑧𝐵 𝑦 = (𝑧 · 𝑥)} ∈ V
2321, 22eqeltri 2886 . . . . . 6 {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V
2423a1i 11 . . . . 5 (𝑥𝐵 → {𝑦 ∣ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V)
2518, 24opabex3 7660 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ∈ V
2616, 17, 25fvmpt 6752 . . 3 (𝑅 ∈ V → (∥r𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
271, 26syl5eq 2845 . 2 (𝑅 ∈ V → = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
28 fvprc 6644 . . . 4 𝑅 ∈ V → (∥r𝑅) = ∅)
291, 28syl5eq 2845 . . 3 𝑅 ∈ V → = ∅)
30 opabn0 5408 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ ↔ ∃𝑥𝑦(𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦))
31 n0i 4251 . . . . . . . 8 (𝑥𝐵 → ¬ 𝐵 = ∅)
32 fvprc 6644 . . . . . . . . 9 𝑅 ∈ V → (Base‘𝑅) = ∅)
333, 32syl5eq 2845 . . . . . . . 8 𝑅 ∈ V → 𝐵 = ∅)
3431, 33nsyl2 143 . . . . . . 7 (𝑥𝐵𝑅 ∈ V)
3534adantr 484 . . . . . 6 ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
3635exlimivv 1933 . . . . 5 (∃𝑥𝑦(𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
3730, 36sylbi 220 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ → 𝑅 ∈ V)
3837necon1bi 3015 . . 3 𝑅 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} = ∅)
3929, 38eqtr4d 2836 . 2 𝑅 ∈ V → = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
4027, 39pm2.61i 185 1 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∃wrex 3107  Vcvv 3441  ∅c0 4245  {copab 5095  ‘cfv 6329  (class class class)co 7142  Basecbs 16492  .rcmulr 16575  ∥rcdsr 19402 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-ov 7145  df-dvdsr 19405 This theorem is referenced by:  dvdsr  19410  dvdsrpropd  19460  dvdsrzring  20194
 Copyright terms: Public domain W3C validator