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Theorem cdj3lem3 25864
Description: Lemma for cdj3i 25867. 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 3564 . . . 4  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21eqeq1i 2450 . . 3  |-  ( ( A  i^i  B )  =  0H  <->  ( B  i^i  A )  =  0H )
3 cdj3lem2.2 . . . . . . . 8  |-  B  e.  SH
43sheli 24638 . . . . . . 7  |-  ( D  e.  B  ->  D  e.  ~H )
5 cdj3lem2.1 . . . . . . . 8  |-  A  e.  SH
65sheli 24638 . . . . . . 7  |-  ( C  e.  A  ->  C  e.  ~H )
7 ax-hvcom 24425 . . . . . . 7  |-  ( ( D  e.  ~H  /\  C  e.  ~H )  ->  ( D  +h  C
)  =  ( C  +h  D ) )
84, 6, 7syl2an 477 . . . . . 6  |-  ( ( D  e.  B  /\  C  e.  A )  ->  ( D  +h  C
)  =  ( C  +h  D ) )
98fveq2d 5716 . . . . 5  |-  ( ( D  e.  B  /\  C  e.  A )  ->  ( T `  ( D  +h  C ) )  =  ( T `  ( C  +h  D
) ) )
1093adant3 1008 . . . 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 24788 . . . . . . 7  |-  ( B  +H  A )  =  ( A  +H  B
)
133sheli 24638 . . . . . . . . . . 11  |-  ( w  e.  B  ->  w  e.  ~H )
145sheli 24638 . . . . . . . . . . 11  |-  ( z  e.  A  ->  z  e.  ~H )
15 ax-hvcom 24425 . . . . . . . . . . 11  |-  ( ( w  e.  ~H  /\  z  e.  ~H )  ->  ( w  +h  z
)  =  ( z  +h  w ) )
1613, 14, 15syl2an 477 . . . . . . . . . 10  |-  ( ( w  e.  B  /\  z  e.  A )  ->  ( w  +h  z
)  =  ( z  +h  w ) )
1716eqeq2d 2454 . . . . . . . . 9  |-  ( ( w  e.  B  /\  z  e.  A )  ->  ( x  =  ( w  +h  z )  <-> 
x  =  ( z  +h  w ) ) )
1817rexbidva 2753 . . . . . . . 8  |-  ( w  e.  B  ->  ( E. z  e.  A  x  =  ( w  +h  z )  <->  E. z  e.  A  x  =  ( z  +h  w
) ) )
1918riotabiia 6091 . . . . . . 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 4397 . . . . . 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 2466 . . . . 5  |-  T  =  ( x  e.  ( B  +H  A ) 
|->  ( iota_ w  e.  B  E. z  e.  A  x  =  ( w  +h  z ) ) )
223, 5, 21cdj3lem2 25861 . . . 4  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( B  i^i  A )  =  0H )  -> 
( T `  ( D  +h  C ) )  =  D )
2310, 22eqtr3d 2477 . . 3  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( B  i^i  A )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
242, 23syl3an3b 1256 . 2  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( A  i^i  B )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
25243com12 1191 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2737    i^i cin 3348    e. cmpt 4371   ` cfv 5439   iota_crio 6072  (class class class)co 6112   ~Hchil 24343    +h cva 24344   SHcsh 24352    +H cph 24355   0Hc0h 24359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-hilex 24423  ax-hfvadd 24424  ax-hvcom 24425  ax-hvass 24426  ax-hv0cl 24427  ax-hvaddid 24428  ax-hfvmul 24429  ax-hvmulid 24430  ax-hvmulass 24431  ax-hvdistr1 24432  ax-hvdistr2 24433  ax-hvmul0 24434
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-grpo 23700  df-ablo 23791  df-hvsub 24395  df-sh 24631  df-ch0 24678  df-shs 24733
This theorem is referenced by:  cdj3lem3a  25865  cdj3i  25867
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