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Theorem cdj3lem3a 25994
Description: Lemma for cdj3i 25996. 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 24870 . . 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 25993 . . . . . . . . 9  |-  ( ( v  e.  A  /\  u  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( T `  (
v  +h  u ) )  =  u )
6 simp2 989 . . . . . . . . 9  |-  ( ( v  e.  A  /\  u  e.  B  /\  ( A  i^i  B )  =  0H )  ->  u  e.  B )
75, 6eqeltrd 2542 . . . . . . . 8  |-  ( ( v  e.  A  /\  u  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( T `  (
v  +h  u ) )  e.  B )
873expa 1188 . . . . . . 7  |-  ( ( ( v  e.  A  /\  u  e.  B
)  /\  ( A  i^i  B )  =  0H )  ->  ( T `  ( v  +h  u
) )  e.  B
)
9 fveq2 5798 . . . . . . . 8  |-  ( C  =  ( v  +h  u )  ->  ( T `  C )  =  ( T `  ( v  +h  u
) ) )
109eleq1d 2523 . . . . . . 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 436 . . . . 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 2951 . . 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 430 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2799    i^i cin 3434    |-> cmpt 4457   ` cfv 5525   iota_crio 6159  (class class class)co 6199    +h cva 24473   SHcsh 24481    +H cph 24484   0Hc0h 24488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-hilex 24552  ax-hfvadd 24553  ax-hvcom 24554  ax-hvass 24555  ax-hv0cl 24556  ax-hvaddid 24557  ax-hfvmul 24558  ax-hvmulid 24559  ax-hvmulass 24560  ax-hvdistr1 24561  ax-hvdistr2 24562  ax-hvmul0 24563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-po 4748  df-so 4749  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-grpo 23829  df-ablo 23920  df-hvsub 24524  df-sh 24760  df-ch0 24807  df-shs 24862
This theorem is referenced by: (None)
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