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

Theorem mdbr 30084
 Description: Binary relation expressing ⟨𝐴, 𝐵⟩ is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
mdbr ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem mdbr
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2877 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 632 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 oveq2 7143 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥 𝑦) = (𝑥 𝐴))
43ineq1d 4138 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥 𝑦) ∩ 𝑧) = ((𝑥 𝐴) ∩ 𝑧))
5 ineq1 4131 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑧) = (𝐴𝑧))
65oveq2d 7151 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 (𝑦𝑧)) = (𝑥 (𝐴𝑧)))
74, 6eqeq12d 2814 . . . . . 6 (𝑦 = 𝐴 → (((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧)) ↔ ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))))
87imbi2d 344 . . . . 5 (𝑦 = 𝐴 → ((𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))) ↔ (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))))
98ralbidv 3162 . . . 4 (𝑦 = 𝐴 → (∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))) ↔ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))))
102, 9anbi12d 633 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧)))) ↔ ((𝐴C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))))))
11 eleq1 2877 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1211anbi2d 631 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
13 sseq2 3941 . . . . . 6 (𝑧 = 𝐵 → (𝑥𝑧𝑥𝐵))
14 ineq2 4133 . . . . . . 7 (𝑧 = 𝐵 → ((𝑥 𝐴) ∩ 𝑧) = ((𝑥 𝐴) ∩ 𝐵))
15 ineq2 4133 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴𝑧) = (𝐴𝐵))
1615oveq2d 7151 . . . . . . 7 (𝑧 = 𝐵 → (𝑥 (𝐴𝑧)) = (𝑥 (𝐴𝐵)))
1714, 16eqeq12d 2814 . . . . . 6 (𝑧 = 𝐵 → (((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)) ↔ ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))
1813, 17imbi12d 348 . . . . 5 (𝑧 = 𝐵 → ((𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))) ↔ (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
1918ralbidv 3162 . . . 4 (𝑧 = 𝐵 → (∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))) ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
2012, 19anbi12d 633 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))) ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))))
21 df-md 30070 . . 3 𝑀 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))))}
2210, 20, 21brabg 5391 . 2 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))))
2322bianabs 545 1 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∩ cin 3880   ⊆ wss 3881   class class class wbr 5030  (class class class)co 7135   Cℋ cch 28719   ∨ℋ chj 28723   𝑀ℋ cmd 28756 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-sep 5167  ax-nul 5174  ax-pr 5295 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-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-iota 6283  df-fv 6332  df-ov 7138  df-md 30070 This theorem is referenced by:  mdi  30085  mdbr2  30086  mdbr3  30087  dmdmd  30090  mddmd2  30099  mdsl1i  30111
 Copyright terms: Public domain W3C validator