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Theorem cdj3lem3 23941
Description: Lemma for cdj3i 23944. Value of the second-component function  T. (Contributed by NM, 23-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
cdj3lem3  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
Distinct variable groups:    x, z, w, A    x, B, z, w    x, C, z, w    x, D, z, w
Allowed substitution hints:    T( x, z, w)

Proof of Theorem cdj3lem3
StepHypRef Expression
1 incom 3533 . . . 4  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21eqeq1i 2443 . . 3  |-  ( ( A  i^i  B )  =  0H  <->  ( B  i^i  A )  =  0H )
3 cdj3lem2.2 . . . . . . . 8  |-  B  e.  SH
43sheli 22716 . . . . . . 7  |-  ( D  e.  B  ->  D  e.  ~H )
5 cdj3lem2.1 . . . . . . . 8  |-  A  e.  SH
65sheli 22716 . . . . . . 7  |-  ( C  e.  A  ->  C  e.  ~H )
7 ax-hvcom 22504 . . . . . . 7  |-  ( ( D  e.  ~H  /\  C  e.  ~H )  ->  ( D  +h  C
)  =  ( C  +h  D ) )
84, 6, 7syl2an 464 . . . . . 6  |-  ( ( D  e.  B  /\  C  e.  A )  ->  ( D  +h  C
)  =  ( C  +h  D ) )
98fveq2d 5732 . . . . 5  |-  ( ( D  e.  B  /\  C  e.  A )  ->  ( T `  ( D  +h  C ) )  =  ( T `  ( C  +h  D
) ) )
1093adant3 977 . . . 4  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( B  i^i  A )  =  0H )  -> 
( T `  ( D  +h  C ) )  =  ( T `  ( C  +h  D
) ) )
11 cdj3lem3.3 . . . . . 6  |-  T  =  ( x  e.  ( A  +H  B ) 
|->  ( iota_ w  e.  B E. z  e.  A  x  =  ( z  +h  w ) ) )
123, 5shscomi 22865 . . . . . . 7  |-  ( B  +H  A )  =  ( A  +H  B
)
133sheli 22716 . . . . . . . . . . 11  |-  ( w  e.  B  ->  w  e.  ~H )
145sheli 22716 . . . . . . . . . . 11  |-  ( z  e.  A  ->  z  e.  ~H )
15 ax-hvcom 22504 . . . . . . . . . . 11  |-  ( ( w  e.  ~H  /\  z  e.  ~H )  ->  ( w  +h  z
)  =  ( z  +h  w ) )
1613, 14, 15syl2an 464 . . . . . . . . . 10  |-  ( ( w  e.  B  /\  z  e.  A )  ->  ( w  +h  z
)  =  ( z  +h  w ) )
1716eqeq2d 2447 . . . . . . . . 9  |-  ( ( w  e.  B  /\  z  e.  A )  ->  ( x  =  ( w  +h  z )  <-> 
x  =  ( z  +h  w ) ) )
1817rexbidva 2722 . . . . . . . 8  |-  ( w  e.  B  ->  ( E. z  e.  A  x  =  ( w  +h  z )  <->  E. z  e.  A  x  =  ( z  +h  w
) ) )
1918riotabiia 6567 . . . . . . 7  |-  ( iota_ w  e.  B E. z  e.  A  x  =  ( w  +h  z
) )  =  (
iota_ w  e.  B E. z  e.  A  x  =  ( z  +h  w ) )
2012, 19mpteq12i 4293 . . . . . 6  |-  ( x  e.  ( B  +H  A )  |->  ( iota_ w  e.  B E. z  e.  A  x  =  ( w  +h  z
) ) )  =  ( x  e.  ( A  +H  B ) 
|->  ( iota_ w  e.  B E. z  e.  A  x  =  ( z  +h  w ) ) )
2111, 20eqtr4i 2459 . . . . 5  |-  T  =  ( x  e.  ( B  +H  A ) 
|->  ( iota_ w  e.  B E. z  e.  A  x  =  ( w  +h  z ) ) )
223, 5, 21cdj3lem2 23938 . . . 4  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( B  i^i  A )  =  0H )  -> 
( T `  ( D  +h  C ) )  =  D )
2310, 22eqtr3d 2470 . . 3  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( B  i^i  A )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
242, 23syl3an3b 1222 . 2  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( A  i^i  B )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
25243com12 1157 1  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706    i^i cin 3319    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   iota_crio 6542   ~Hchil 22422    +h cva 22423   SHcsh 22431    +H cph 22434   0Hc0h 22438
This theorem is referenced by:  cdj3lem3a  23942  cdj3i  23944
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-hilex 22502  ax-hfvadd 22503  ax-hvcom 22504  ax-hvass 22505  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvmulass 22510  ax-hvdistr1 22511  ax-hvdistr2 22512  ax-hvmul0 22513
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-grpo 21779  df-ablo 21870  df-hvsub 22474  df-sh 22709  df-ch0 22755  df-shs 22810
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