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Theorem cdj3lem3 27770
Description: Lemma for cdj3i 27773. 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 3632 . . . 4  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21eqeq1i 2409 . . 3  |-  ( ( A  i^i  B )  =  0H  <->  ( B  i^i  A )  =  0H )
3 cdj3lem2.2 . . . . . . . 8  |-  B  e.  SH
43sheli 26545 . . . . . . 7  |-  ( D  e.  B  ->  D  e.  ~H )
5 cdj3lem2.1 . . . . . . . 8  |-  A  e.  SH
65sheli 26545 . . . . . . 7  |-  ( C  e.  A  ->  C  e.  ~H )
7 ax-hvcom 26332 . . . . . . 7  |-  ( ( D  e.  ~H  /\  C  e.  ~H )  ->  ( D  +h  C
)  =  ( C  +h  D ) )
84, 6, 7syl2an 475 . . . . . 6  |-  ( ( D  e.  B  /\  C  e.  A )  ->  ( D  +h  C
)  =  ( C  +h  D ) )
98fveq2d 5853 . . . . 5  |-  ( ( D  e.  B  /\  C  e.  A )  ->  ( T `  ( D  +h  C ) )  =  ( T `  ( C  +h  D
) ) )
1093adant3 1017 . . . 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 26695 . . . . . . 7  |-  ( B  +H  A )  =  ( A  +H  B
)
133sheli 26545 . . . . . . . . . . 11  |-  ( w  e.  B  ->  w  e.  ~H )
145sheli 26545 . . . . . . . . . . 11  |-  ( z  e.  A  ->  z  e.  ~H )
15 ax-hvcom 26332 . . . . . . . . . . 11  |-  ( ( w  e.  ~H  /\  z  e.  ~H )  ->  ( w  +h  z
)  =  ( z  +h  w ) )
1613, 14, 15syl2an 475 . . . . . . . . . 10  |-  ( ( w  e.  B  /\  z  e.  A )  ->  ( w  +h  z
)  =  ( z  +h  w ) )
1716eqeq2d 2416 . . . . . . . . 9  |-  ( ( w  e.  B  /\  z  e.  A )  ->  ( x  =  ( w  +h  z )  <-> 
x  =  ( z  +h  w ) ) )
1817rexbidva 2915 . . . . . . . 8  |-  ( w  e.  B  ->  ( E. z  e.  A  x  =  ( w  +h  z )  <->  E. z  e.  A  x  =  ( z  +h  w
) ) )
1918riotabiia 6257 . . . . . . 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 4479 . . . . . 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 2434 . . . . 5  |-  T  =  ( x  e.  ( B  +H  A ) 
|->  ( iota_ w  e.  B  E. z  e.  A  x  =  ( w  +h  z ) ) )
223, 5, 21cdj3lem2 27767 . . . 4  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( B  i^i  A )  =  0H )  -> 
( T `  ( D  +h  C ) )  =  D )
2310, 22eqtr3d 2445 . . 3  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( B  i^i  A )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
242, 23syl3an3b 1268 . 2  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( A  i^i  B )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
25243com12 1201 1  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   E.wrex 2755    i^i cin 3413    |-> cmpt 4453   ` cfv 5569   iota_crio 6239  (class class class)co 6278   ~Hchil 26250    +h cva 26251   SHcsh 26259    +H cph 26262   0Hc0h 26266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-hilex 26330  ax-hfvadd 26331  ax-hvcom 26332  ax-hvass 26333  ax-hv0cl 26334  ax-hvaddid 26335  ax-hfvmul 26336  ax-hvmulid 26337  ax-hvmulass 26338  ax-hvdistr1 26339  ax-hvdistr2 26340  ax-hvmul0 26341
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-grpo 25607  df-ablo 25698  df-hvsub 26302  df-sh 26538  df-ch0 26585  df-shs 26640
This theorem is referenced by:  cdj3lem3a  27771  cdj3i  27773
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