Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  dmdbr Structured version   Visualization version   GIF version

Theorem dmdbr 30085
 Description: Binary relation expressing the dual modular pair property. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
dmdbr ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dmdbr
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2880 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 632 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 ineq2 4136 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
43oveq1d 7154 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥𝑦) ∨ 𝑧) = ((𝑥𝐴) ∨ 𝑧))
5 oveq1 7146 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦 𝑧) = (𝐴 𝑧))
65ineq2d 4142 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 ∩ (𝑦 𝑧)) = (𝑥 ∩ (𝐴 𝑧)))
74, 6eqeq12d 2817 . . . . . 6 (𝑦 = 𝐴 → (((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧)) ↔ ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))))
87imbi2d 344 . . . . 5 (𝑦 = 𝐴 → ((𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))) ↔ (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))))
98ralbidv 3165 . . . 4 (𝑦 = 𝐴 → (∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))) ↔ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))))
102, 9anbi12d 633 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧)))) ↔ ((𝐴C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))))))
11 eleq1 2880 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1211anbi2d 631 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
13 sseq1 3943 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑥𝐵𝑥))
14 oveq2 7147 . . . . . . 7 (𝑧 = 𝐵 → ((𝑥𝐴) ∨ 𝑧) = ((𝑥𝐴) ∨ 𝐵))
15 oveq2 7147 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴 𝑧) = (𝐴 𝐵))
1615ineq2d 4142 . . . . . . 7 (𝑧 = 𝐵 → (𝑥 ∩ (𝐴 𝑧)) = (𝑥 ∩ (𝐴 𝐵)))
1714, 16eqeq12d 2817 . . . . . 6 (𝑧 = 𝐵 → (((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)) ↔ ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))
1813, 17imbi12d 348 . . . . 5 (𝑧 = 𝐵 → ((𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))) ↔ (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
1918ralbidv 3165 . . . 4 (𝑧 = 𝐵 → (∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))) ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
2012, 19anbi12d 633 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))) ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))))
21 df-dmd 30067 . . 3 𝑀* = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))))}
2210, 20, 21brabg 5394 . 2 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))))
2322bianabs 545 1 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∀wral 3109   ∩ cin 3883   ⊆ wss 3884   class class class wbr 5033  (class class class)co 7139   Cℋ cch 28715   ∨ℋ chj 28719   𝑀ℋ* cdmd 28753 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-iota 6287  df-fv 6336  df-ov 7142  df-dmd 30067 This theorem is referenced by:  dmdmd  30086  dmdi  30088  dmdbr2  30089  dmdbr3  30091  mddmd2  30095
 Copyright terms: Public domain W3C validator