HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  mdi Structured version   Unicode version

Theorem mdi 25699
Description: Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
mdi  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH  B  /\  C  C_  B ) )  ->  ( ( C  vH  A )  i^i 
B )  =  ( C  vH  ( A  i^i  B ) ) )

Proof of Theorem mdi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mdbr 25698 . . . . 5  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
21biimpd 207 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  ->  A. x  e.  CH  ( x  C_  B  -> 
( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) ) )
3 sseq1 3377 . . . . . 6  |-  ( x  =  C  ->  (
x  C_  B  <->  C  C_  B
) )
4 oveq1 6098 . . . . . . . 8  |-  ( x  =  C  ->  (
x  vH  A )  =  ( C  vH  A ) )
54ineq1d 3551 . . . . . . 7  |-  ( x  =  C  ->  (
( x  vH  A
)  i^i  B )  =  ( ( C  vH  A )  i^i 
B ) )
6 oveq1 6098 . . . . . . 7  |-  ( x  =  C  ->  (
x  vH  ( A  i^i  B ) )  =  ( C  vH  ( A  i^i  B ) ) )
75, 6eqeq12d 2457 . . . . . 6  |-  ( x  =  C  ->  (
( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) )  <->  ( ( C  vH  A )  i^i 
B )  =  ( C  vH  ( A  i^i  B ) ) ) )
83, 7imbi12d 320 . . . . 5  |-  ( x  =  C  ->  (
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) )  <->  ( C  C_  B  ->  ( ( C  vH  A )  i^i 
B )  =  ( C  vH  ( A  i^i  B ) ) ) ) )
98rspcv 3069 . . . 4  |-  ( C  e.  CH  ->  ( A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )  ->  ( C  C_  B  ->  (
( C  vH  A
)  i^i  B )  =  ( C  vH  ( A  i^i  B ) ) ) ) )
102, 9sylan9 657 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  C  e.  CH )  ->  ( A  MH  B  ->  ( C  C_  B  ->  ( ( C  vH  A )  i^i  B
)  =  ( C  vH  ( A  i^i  B ) ) ) ) )
11103impa 1182 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  MH  B  ->  ( C  C_  B  ->  ( ( C  vH  A
)  i^i  B )  =  ( C  vH  ( A  i^i  B ) ) ) ) )
1211imp32 433 1  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  MH  B  /\  C  C_  B ) )  ->  ( ( C  vH  A )  i^i 
B )  =  ( C  vH  ( A  i^i  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715    i^i cin 3327    C_ wss 3328   class class class wbr 4292  (class class class)co 6091   CHcch 24331    vH chj 24335    MH cmd 24368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-iota 5381  df-fv 5426  df-ov 6094  df-md 25684
This theorem is referenced by:  mdsl3  25720  mdslmd3i  25736  mdexchi  25739  atabsi  25805
  Copyright terms: Public domain W3C validator