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

Theorem mdbr 29985
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 2905 . . . . 5 (𝑦 = 𝐴 → (𝑦C𝐴C ))
21anbi1d 629 . . . 4 (𝑦 = 𝐴 → ((𝑦C𝑧C ) ↔ (𝐴C𝑧C )))
3 oveq2 7156 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥 𝑦) = (𝑥 𝐴))
43ineq1d 4192 . . . . . . 7 (𝑦 = 𝐴 → ((𝑥 𝑦) ∩ 𝑧) = ((𝑥 𝐴) ∩ 𝑧))
5 ineq1 4185 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑧) = (𝐴𝑧))
65oveq2d 7164 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 (𝑦𝑧)) = (𝑥 (𝐴𝑧)))
74, 6eqeq12d 2842 . . . . . 6 (𝑦 = 𝐴 → (((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧)) ↔ ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))))
87imbi2d 342 . . . . 5 (𝑦 = 𝐴 → ((𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))) ↔ (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))))
98ralbidv 3202 . . . 4 (𝑦 = 𝐴 → (∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))) ↔ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))))
102, 9anbi12d 630 . . 3 (𝑦 = 𝐴 → (((𝑦C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧)))) ↔ ((𝐴C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))))))
11 eleq1 2905 . . . . 5 (𝑧 = 𝐵 → (𝑧C𝐵C ))
1211anbi2d 628 . . . 4 (𝑧 = 𝐵 → ((𝐴C𝑧C ) ↔ (𝐴C𝐵C )))
13 sseq2 3997 . . . . . 6 (𝑧 = 𝐵 → (𝑥𝑧𝑥𝐵))
14 ineq2 4187 . . . . . . 7 (𝑧 = 𝐵 → ((𝑥 𝐴) ∩ 𝑧) = ((𝑥 𝐴) ∩ 𝐵))
15 ineq2 4187 . . . . . . . 8 (𝑧 = 𝐵 → (𝐴𝑧) = (𝐴𝐵))
1615oveq2d 7164 . . . . . . 7 (𝑧 = 𝐵 → (𝑥 (𝐴𝑧)) = (𝑥 (𝐴𝐵)))
1714, 16eqeq12d 2842 . . . . . 6 (𝑧 = 𝐵 → (((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)) ↔ ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))
1813, 17imbi12d 346 . . . . 5 (𝑧 = 𝐵 → ((𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))) ↔ (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
1918ralbidv 3202 . . . 4 (𝑧 = 𝐵 → (∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧))) ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
2012, 19anbi12d 630 . . 3 (𝑧 = 𝐵 → (((𝐴C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝐴) ∩ 𝑧) = (𝑥 (𝐴𝑧)))) ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))))
21 df-md 29971 . . 3 𝑀 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦C𝑧C ) ∧ ∀𝑥C (𝑥𝑧 → ((𝑥 𝑦) ∩ 𝑧) = (𝑥 (𝑦𝑧))))}
2210, 20, 21brabg 5423 . 2 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ((𝐴C𝐵C ) ∧ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵))))))
2322bianabs 542 1 ((𝐴C𝐵C ) → (𝐴 𝑀 𝐵 ↔ ∀𝑥C (𝑥𝐵 → ((𝑥 𝐴) ∩ 𝐵) = (𝑥 (𝐴𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  cin 3939  wss 3940   class class class wbr 5063  (class class class)co 7148   C cch 28620   chj 28624   𝑀 cmd 28657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-iota 6312  df-fv 6360  df-ov 7151  df-md 29971
This theorem is referenced by:  mdi  29986  mdbr2  29987  mdbr3  29988  dmdmd  29991  mddmd2  30000  mdsl1i  30012
  Copyright terms: Public domain W3C validator