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

Theorem dmdbr 31283
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 2826 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 631 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 ineq2 4171 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
43oveq1d 7377 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥𝑦) ∨ 𝑧) = ((𝑥𝐴) ∨ 𝑧))
5 oveq1 7369 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦 𝑧) = (𝐴 𝑧))
65ineq2d 4177 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 ∩ (𝑦 𝑧)) = (𝑥 ∩ (𝐴 𝑧)))
74, 6eqeq12d 2753 . . . . . 6 (𝑦 = 𝐴 → (((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧)) ↔ ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))))
87imbi2d 341 . . . . 5 (𝑦 = 𝐴 → ((𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))) ↔ (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))))
98ralbidv 3175 . . . 4 (𝑦 = 𝐴 → (∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))) ↔ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))))
102, 9anbi12d 632 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧)))) ↔ ((𝐴C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))))))
11 eleq1 2826 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1211anbi2d 630 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
13 sseq1 3974 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑥𝐵𝑥))
14 oveq2 7370 . . . . . . 7 (𝑧 = 𝐵 → ((𝑥𝐴) ∨ 𝑧) = ((𝑥𝐴) ∨ 𝐵))
15 oveq2 7370 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴 𝑧) = (𝐴 𝐵))
1615ineq2d 4177 . . . . . . 7 (𝑧 = 𝐵 → (𝑥 ∩ (𝐴 𝑧)) = (𝑥 ∩ (𝐴 𝐵)))
1714, 16eqeq12d 2753 . . . . . 6 (𝑧 = 𝐵 → (((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)) ↔ ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))
1813, 17imbi12d 345 . . . . 5 (𝑧 = 𝐵 → ((𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))) ↔ (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
1918ralbidv 3175 . . . 4 (𝑧 = 𝐵 → (∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧))) ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
2012, 19anbi12d 632 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝐴) ∨ 𝑧) = (𝑥 ∩ (𝐴 𝑧)))) ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))))
21 df-dmd 31265 . . 3 𝑀* = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ ∀𝑥C (𝑧𝑥 → ((𝑥𝑦) ∨ 𝑧) = (𝑥 ∩ (𝑦 𝑧))))}
2210, 20, 21brabg 5501 . 2 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵))))))
2322bianabs 543 1 ((𝐴C𝐵C ) → (𝐴 𝑀* 𝐵 ↔ ∀𝑥C (𝐵𝑥 → ((𝑥𝐴) ∨ 𝐵) = (𝑥 ∩ (𝐴 𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  cin 3914  wss 3915   class class class wbr 5110  (class class class)co 7362   C cch 29913   chj 29917   𝑀* cdmd 29951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-iota 6453  df-fv 6509  df-ov 7365  df-dmd 31265
This theorem is referenced by:  dmdmd  31284  dmdi  31286  dmdbr2  31287  dmdbr3  31289  mddmd2  31293
  Copyright terms: Public domain W3C validator