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Theorem cdj3lem3 23018
Description: Lemma for cdj3i 23021. 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 3361 . . . 4  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21eqeq1i 2290 . . 3  |-  ( ( A  i^i  B )  =  0H  <->  ( B  i^i  A )  =  0H )
3 cdj3lem2.2 . . . . . . . 8  |-  B  e.  SH
43sheli 21793 . . . . . . 7  |-  ( D  e.  B  ->  D  e.  ~H )
5 cdj3lem2.1 . . . . . . . 8  |-  A  e.  SH
65sheli 21793 . . . . . . 7  |-  ( C  e.  A  ->  C  e.  ~H )
7 ax-hvcom 21581 . . . . . . 7  |-  ( ( D  e.  ~H  /\  C  e.  ~H )  ->  ( D  +h  C
)  =  ( C  +h  D ) )
84, 6, 7syl2an 463 . . . . . 6  |-  ( ( D  e.  B  /\  C  e.  A )  ->  ( D  +h  C
)  =  ( C  +h  D ) )
98fveq2d 5529 . . . . 5  |-  ( ( D  e.  B  /\  C  e.  A )  ->  ( T `  ( D  +h  C ) )  =  ( T `  ( C  +h  D
) ) )
1093adant3 975 . . . 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 21942 . . . . . . 7  |-  ( B  +H  A )  =  ( A  +H  B
)
133sheli 21793 . . . . . . . . . . 11  |-  ( w  e.  B  ->  w  e.  ~H )
145sheli 21793 . . . . . . . . . . 11  |-  ( z  e.  A  ->  z  e.  ~H )
15 ax-hvcom 21581 . . . . . . . . . . 11  |-  ( ( w  e.  ~H  /\  z  e.  ~H )  ->  ( w  +h  z
)  =  ( z  +h  w ) )
1613, 14, 15syl2an 463 . . . . . . . . . 10  |-  ( ( w  e.  B  /\  z  e.  A )  ->  ( w  +h  z
)  =  ( z  +h  w ) )
1716eqeq2d 2294 . . . . . . . . 9  |-  ( ( w  e.  B  /\  z  e.  A )  ->  ( x  =  ( w  +h  z )  <-> 
x  =  ( z  +h  w ) ) )
1817rexbidva 2560 . . . . . . . 8  |-  ( w  e.  B  ->  ( E. z  e.  A  x  =  ( w  +h  z )  <->  E. z  e.  A  x  =  ( z  +h  w
) ) )
1918riotabiia 6322 . . . . . . 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 4104 . . . . . 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 2306 . . . . 5  |-  T  =  ( x  e.  ( B  +H  A ) 
|->  ( iota_ w  e.  B E. z  e.  A  x  =  ( w  +h  z ) ) )
223, 5, 21cdj3lem2 23015 . . . 4  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( B  i^i  A )  =  0H )  -> 
( T `  ( D  +h  C ) )  =  D )
2310, 22eqtr3d 2317 . . 3  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( B  i^i  A )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
242, 23syl3an3b 1220 . 2  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( A  i^i  B )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
25243com12 1155 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   ~Hchil 21499    +h cva 21500   SHcsh 21508    +H cph 21511   0Hc0h 21515
This theorem is referenced by:  cdj3lem3a  23019  cdj3i  23021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-grpo 20858  df-ablo 20949  df-hvsub 21551  df-sh 21786  df-ch0 21832  df-shs 21887
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