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Theorem cdj3lem2a 25993
Description: Lemma for cdj3i 25998. Closure of the first-component function  S. (Contributed by NM, 25-May-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdj3lem2.1  |-  A  e.  SH
cdj3lem2.2  |-  B  e.  SH
cdj3lem2.3  |-  S  =  ( x  e.  ( A  +H  B ) 
|->  ( iota_ z  e.  A  E. w  e.  B  x  =  ( z  +h  w ) ) )
Assertion
Ref Expression
cdj3lem2a  |-  ( ( C  e.  ( A  +H  B )  /\  ( A  i^i  B )  =  0H )  -> 
( S `  C
)  e.  A )
Distinct variable groups:    x, z, w, A    x, B, z, w    x, C, z, w
Allowed substitution hints:    S( x, z, w)

Proof of Theorem cdj3lem2a
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 24872 . . 3  |-  ( C  e.  ( A  +H  B )  <->  E. v  e.  A  E. u  e.  B  C  =  ( v  +h  u
) )
4 cdj3lem2.3 . . . . . . . . . 10  |-  S  =  ( x  e.  ( A  +H  B ) 
|->  ( iota_ z  e.  A  E. w  e.  B  x  =  ( z  +h  w ) ) )
51, 2, 4cdj3lem2 25992 . . . . . . . . 9  |-  ( ( v  e.  A  /\  u  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( S `  (
v  +h  u ) )  =  v )
6 simp1 988 . . . . . . . . 9  |-  ( ( v  e.  A  /\  u  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
v  e.  A )
75, 6eqeltrd 2542 . . . . . . . 8  |-  ( ( v  e.  A  /\  u  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( S `  (
v  +h  u ) )  e.  A )
873expa 1188 . . . . . . 7  |-  ( ( ( v  e.  A  /\  u  e.  B
)  /\  ( A  i^i  B )  =  0H )  ->  ( S `  ( v  +h  u
) )  e.  A
)
9 fveq2 5800 . . . . . . . 8  |-  ( C  =  ( v  +h  u )  ->  ( S `  C )  =  ( S `  ( v  +h  u
) ) )
109eleq1d 2523 . . . . . . 7  |-  ( C  =  ( v  +h  u )  ->  (
( S `  C
)  e.  A  <->  ( S `  ( v  +h  u
) )  e.  A
) )
118, 10syl5ibr 221 . . . . . 6  |-  ( C  =  ( v  +h  u )  ->  (
( ( v  e.  A  /\  u  e.  B )  /\  ( A  i^i  B )  =  0H )  ->  ( S `  C )  e.  A ) )
1211expd 436 . . . . 5  |-  ( C  =  ( v  +h  u )  ->  (
( v  e.  A  /\  u  e.  B
)  ->  ( ( A  i^i  B )  =  0H  ->  ( S `  C )  e.  A
) ) )
1312com13 80 . . . 4  |-  ( ( A  i^i  B )  =  0H  ->  (
( v  e.  A  /\  u  e.  B
)  ->  ( C  =  ( v  +h  u )  ->  ( S `  C )  e.  A ) ) )
1413rexlimdvv 2953 . . 3  |-  ( ( A  i^i  B )  =  0H  ->  ( E. v  e.  A  E. u  e.  B  C  =  ( v  +h  u )  ->  ( S `  C )  e.  A ) )
153, 14syl5bi 217 . 2  |-  ( ( A  i^i  B )  =  0H  ->  ( C  e.  ( A  +H  B )  ->  ( S `  C )  e.  A ) )
1615impcom 430 1  |-  ( ( C  e.  ( A  +H  B )  /\  ( A  i^i  B )  =  0H )  -> 
( S `  C
)  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2800    i^i cin 3436    |-> cmpt 4459   ` cfv 5527   iota_crio 6161  (class class class)co 6201    +h cva 24475   SHcsh 24483    +H cph 24486   0Hc0h 24490
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 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-hilex 24554  ax-hfvadd 24555  ax-hvcom 24556  ax-hvass 24557  ax-hv0cl 24558  ax-hvaddid 24559  ax-hfvmul 24560  ax-hvmulid 24561  ax-hvmulass 24562  ax-hvdistr1 24563  ax-hvdistr2 24564  ax-hvmul0 24565
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 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-grpo 23831  df-ablo 23922  df-hvsub 24526  df-sh 24762  df-ch0 24809  df-shs 24864
This theorem is referenced by: (None)
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