HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  cdj3lem3a Structured version   Unicode version

Theorem cdj3lem3a 27474
Description: Lemma for cdj3i 27476. Closure of the second-component function  T. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdj3lem2.1  |-  A  e.  SH
cdj3lem2.2  |-  B  e.  SH
cdj3lem3.3  |-  T  =  ( x  e.  ( A  +H  B ) 
|->  ( iota_ w  e.  B  E. z  e.  A  x  =  ( z  +h  w ) ) )
Assertion
Ref Expression
cdj3lem3a  |-  ( ( C  e.  ( A  +H  B )  /\  ( A  i^i  B )  =  0H )  -> 
( T `  C
)  e.  B )
Distinct variable groups:    x, z, w, A    x, B, z, w    x, C, z, w
Allowed substitution hints:    T( x, z, w)

Proof of Theorem cdj3lem3a
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdj3lem2.1 . . . 4  |-  A  e.  SH
2 cdj3lem2.2 . . . 4  |-  B  e.  SH
31, 2shseli 26351 . . 3  |-  ( C  e.  ( A  +H  B )  <->  E. v  e.  A  E. u  e.  B  C  =  ( v  +h  u
) )
4 cdj3lem3.3 . . . . . . . . . 10  |-  T  =  ( x  e.  ( A  +H  B ) 
|->  ( iota_ w  e.  B  E. z  e.  A  x  =  ( z  +h  w ) ) )
51, 2, 4cdj3lem3 27473 . . . . . . . . 9  |-  ( ( v  e.  A  /\  u  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( T `  (
v  +h  u ) )  =  u )
6 simp2 995 . . . . . . . . 9  |-  ( ( v  e.  A  /\  u  e.  B  /\  ( A  i^i  B )  =  0H )  ->  u  e.  B )
75, 6eqeltrd 2470 . . . . . . . 8  |-  ( ( v  e.  A  /\  u  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( T `  (
v  +h  u ) )  e.  B )
873expa 1194 . . . . . . 7  |-  ( ( ( v  e.  A  /\  u  e.  B
)  /\  ( A  i^i  B )  =  0H )  ->  ( T `  ( v  +h  u
) )  e.  B
)
9 fveq2 5774 . . . . . . . 8  |-  ( C  =  ( v  +h  u )  ->  ( T `  C )  =  ( T `  ( v  +h  u
) ) )
109eleq1d 2451 . . . . . . 7  |-  ( C  =  ( v  +h  u )  ->  (
( T `  C
)  e.  B  <->  ( T `  ( v  +h  u
) )  e.  B
) )
118, 10syl5ibr 221 . . . . . 6  |-  ( C  =  ( v  +h  u )  ->  (
( ( v  e.  A  /\  u  e.  B )  /\  ( A  i^i  B )  =  0H )  ->  ( T `  C )  e.  B ) )
1211expd 434 . . . . 5  |-  ( C  =  ( v  +h  u )  ->  (
( v  e.  A  /\  u  e.  B
)  ->  ( ( A  i^i  B )  =  0H  ->  ( T `  C )  e.  B
) ) )
1312com13 80 . . . 4  |-  ( ( A  i^i  B )  =  0H  ->  (
( v  e.  A  /\  u  e.  B
)  ->  ( C  =  ( v  +h  u )  ->  ( T `  C )  e.  B ) ) )
1413rexlimdvv 2880 . . 3  |-  ( ( A  i^i  B )  =  0H  ->  ( E. v  e.  A  E. u  e.  B  C  =  ( v  +h  u )  ->  ( T `  C )  e.  B ) )
153, 14syl5bi 217 . 2  |-  ( ( A  i^i  B )  =  0H  ->  ( C  e.  ( A  +H  B )  ->  ( T `  C )  e.  B ) )
1615impcom 428 1  |-  ( ( C  e.  ( A  +H  B )  /\  ( A  i^i  B )  =  0H )  -> 
( T `  C
)  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   E.wrex 2733    i^i cin 3388    |-> cmpt 4425   ` cfv 5496   iota_crio 6157  (class class class)co 6196    +h cva 25954   SHcsh 25962    +H cph 25965   0Hc0h 25969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-hilex 26033  ax-hfvadd 26034  ax-hvcom 26035  ax-hvass 26036  ax-hv0cl 26037  ax-hvaddid 26038  ax-hfvmul 26039  ax-hvmulid 26040  ax-hvmulass 26041  ax-hvdistr1 26042  ax-hvdistr2 26043  ax-hvmul0 26044
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-grpo 25310  df-ablo 25401  df-hvsub 26005  df-sh 26241  df-ch0 26288  df-shs 26343
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator