HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  mdbr3 Structured version   Unicode version

Theorem mdbr3 23792
Description: Binary relation expressing the modular pair property. This version quantifies an equality instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.)
Assertion
Ref Expression
mdbr3  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem mdbr3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 mdbr 23789 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) ) ) )
2 chincl 22993 . . . . . . . 8  |-  ( ( x  e.  CH  /\  B  e.  CH )  ->  ( x  i^i  B
)  e.  CH )
3 inss2 3554 . . . . . . . . 9  |-  ( x  i^i  B )  C_  B
4 sseq1 3361 . . . . . . . . . . 11  |-  ( y  =  ( x  i^i 
B )  ->  (
y  C_  B  <->  ( x  i^i  B )  C_  B
) )
5 oveq1 6080 . . . . . . . . . . . . 13  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  A )  =  ( ( x  i^i  B )  vH  A ) )
65ineq1d 3533 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
( y  vH  A
)  i^i  B )  =  ( ( ( x  i^i  B )  vH  A )  i^i 
B ) )
7 oveq1 6080 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  ( A  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( A  i^i  B ) ) )
86, 7eqeq12d 2449 . . . . . . . . . . 11  |-  ( y  =  ( x  i^i 
B )  ->  (
( ( y  vH  A )  i^i  B
)  =  ( y  vH  ( A  i^i  B ) )  <->  ( (
( x  i^i  B
)  vH  A )  i^i  B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) ) ) )
94, 8imbi12d 312 . . . . . . . . . 10  |-  ( y  =  ( x  i^i 
B )  ->  (
( y  C_  B  ->  ( ( y  vH  A )  i^i  B
)  =  ( y  vH  ( A  i^i  B ) ) )  <->  ( (
x  i^i  B )  C_  B  ->  ( (
( x  i^i  B
)  vH  A )  i^i  B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) ) ) ) )
109rspcv 3040 . . . . . . . . 9  |-  ( ( x  i^i  B )  e.  CH  ->  ( A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) )  ->  (
( x  i^i  B
)  C_  B  ->  ( ( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) ) )
113, 10mpii 41 . . . . . . . 8  |-  ( ( x  i^i  B )  e.  CH  ->  ( A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) )  ->  (
( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) )
122, 11syl 16 . . . . . . 7  |-  ( ( x  e.  CH  /\  B  e.  CH )  ->  ( A. y  e. 
CH  ( y  C_  B  ->  ( ( y  vH  A )  i^i 
B )  =  ( y  vH  ( A  i^i  B ) ) )  ->  ( (
( x  i^i  B
)  vH  A )  i^i  B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) ) ) )
1312ex 424 . . . . . 6  |-  ( x  e.  CH  ->  ( B  e.  CH  ->  ( A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) )  ->  (
( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) ) )
1413com3l 77 . . . . 5  |-  ( B  e.  CH  ->  ( A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) )  ->  (
x  e.  CH  ->  ( ( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) ) )
1514ralrimdv 2787 . . . 4  |-  ( B  e.  CH  ->  ( A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) )  ->  A. x  e.  CH  ( ( ( x  i^i  B )  vH  A )  i^i 
B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) ) ) )
16 dfss 3327 . . . . . . . . . . 11  |-  ( x 
C_  B  <->  x  =  ( x  i^i  B ) )
1716biimpi 187 . . . . . . . . . 10  |-  ( x 
C_  B  ->  x  =  ( x  i^i 
B ) )
1817oveq1d 6088 . . . . . . . . 9  |-  ( x 
C_  B  ->  (
x  vH  A )  =  ( ( x  i^i  B )  vH  A ) )
1918ineq1d 3533 . . . . . . . 8  |-  ( x 
C_  B  ->  (
( x  vH  A
)  i^i  B )  =  ( ( ( x  i^i  B )  vH  A )  i^i 
B ) )
2017oveq1d 6088 . . . . . . . 8  |-  ( x 
C_  B  ->  (
x  vH  ( A  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( A  i^i  B ) ) )
2119, 20eqeq12d 2449 . . . . . . 7  |-  ( x 
C_  B  ->  (
( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) )  <->  ( (
( x  i^i  B
)  vH  A )  i^i  B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) ) ) )
2221biimprcd 217 . . . . . 6  |-  ( ( ( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) )  ->  ( x  C_  B  ->  ( (
x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) )
2322ralimi 2773 . . . . 5  |-  ( A. x  e.  CH  ( ( ( x  i^i  B
)  vH  A )  i^i  B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) )  ->  A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) )
24 sseq1 3361 . . . . . . 7  |-  ( x  =  y  ->  (
x  C_  B  <->  y  C_  B ) )
25 oveq1 6080 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  vH  A )  =  ( y  vH  A ) )
2625ineq1d 3533 . . . . . . . 8  |-  ( x  =  y  ->  (
( x  vH  A
)  i^i  B )  =  ( ( y  vH  A )  i^i 
B ) )
27 oveq1 6080 . . . . . . . 8  |-  ( x  =  y  ->  (
x  vH  ( A  i^i  B ) )  =  ( y  vH  ( A  i^i  B ) ) )
2826, 27eqeq12d 2449 . . . . . . 7  |-  ( x  =  y  ->  (
( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) )  <->  ( (
y  vH  A )  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) ) )
2924, 28imbi12d 312 . . . . . 6  |-  ( x  =  y  ->  (
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) )  <->  ( y  C_  B  ->  ( (
y  vH  A )  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) ) ) )
3029cbvralv 2924 . . . . 5  |-  ( A. x  e.  CH  ( x 
C_  B  ->  (
( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )  <->  A. y  e.  CH  ( y  C_  B  ->  ( ( y  vH  A )  i^i 
B )  =  ( y  vH  ( A  i^i  B ) ) ) )
3123, 30sylib 189 . . . 4  |-  ( A. x  e.  CH  ( ( ( x  i^i  B
)  vH  A )  i^i  B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) )  ->  A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) ) )
3215, 31impbid1 195 . . 3  |-  ( B  e.  CH  ->  ( A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) )  <->  A. x  e.  CH  ( ( ( x  i^i  B )  vH  A )  i^i 
B )  =  ( ( x  i^i  B
)  vH  ( A  i^i  B ) ) ) )
3332adantl 453 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A. y  e. 
CH  ( y  C_  B  ->  ( ( y  vH  A )  i^i 
B )  =  ( y  vH  ( A  i^i  B ) ) )  <->  A. x  e.  CH  ( ( ( x  i^i  B )  vH  A )  i^i  B
)  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) )
341, 33bitrd 245 1  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
( ( x  i^i 
B )  vH  A
)  i^i  B )  =  ( ( x  i^i  B )  vH  ( A  i^i  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    i^i cin 3311    C_ wss 3312   class class class wbr 4204  (class class class)co 6073   CHcch 22424    vH chj 22428    MH cmd 22461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rrecex 9054  ax-cnre 9055  ax-hilex 22494  ax-hfvadd 22495  ax-hv0cl 22498  ax-hfvmul 22500
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-map 7012  df-nn 9993  df-hlim 22467  df-sh 22701  df-ch 22716  df-md 23775
  Copyright terms: Public domain W3C validator