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Theorem cdj3lem3 27019
Description: Lemma for cdj3i 27022. 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 3684 . . . 4  |-  ( A  i^i  B )  =  ( B  i^i  A
)
21eqeq1i 2467 . . 3  |-  ( ( A  i^i  B )  =  0H  <->  ( B  i^i  A )  =  0H )
3 cdj3lem2.2 . . . . . . . 8  |-  B  e.  SH
43sheli 25793 . . . . . . 7  |-  ( D  e.  B  ->  D  e.  ~H )
5 cdj3lem2.1 . . . . . . . 8  |-  A  e.  SH
65sheli 25793 . . . . . . 7  |-  ( C  e.  A  ->  C  e.  ~H )
7 ax-hvcom 25580 . . . . . . 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 5861 . . . . 5  |-  ( ( D  e.  B  /\  C  e.  A )  ->  ( T `  ( D  +h  C ) )  =  ( T `  ( C  +h  D
) ) )
1093adant3 1011 . . . 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 25943 . . . . . . 7  |-  ( B  +H  A )  =  ( A  +H  B
)
133sheli 25793 . . . . . . . . . . 11  |-  ( w  e.  B  ->  w  e.  ~H )
145sheli 25793 . . . . . . . . . . 11  |-  ( z  e.  A  ->  z  e.  ~H )
15 ax-hvcom 25580 . . . . . . . . . . 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 2474 . . . . . . . . 9  |-  ( ( w  e.  B  /\  z  e.  A )  ->  ( x  =  ( w  +h  z )  <-> 
x  =  ( z  +h  w ) ) )
1817rexbidva 2963 . . . . . . . 8  |-  ( w  e.  B  ->  ( E. z  e.  A  x  =  ( w  +h  z )  <->  E. z  e.  A  x  =  ( z  +h  w
) ) )
1918riotabiia 6254 . . . . . . 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 4524 . . . . . 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 2492 . . . . 5  |-  T  =  ( x  e.  ( B  +H  A ) 
|->  ( iota_ w  e.  B  E. z  e.  A  x  =  ( w  +h  z ) ) )
223, 5, 21cdj3lem2 27016 . . . 4  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( B  i^i  A )  =  0H )  -> 
( T `  ( D  +h  C ) )  =  D )
2310, 22eqtr3d 2503 . . 3  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( B  i^i  A )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
242, 23syl3an3b 1261 . 2  |-  ( ( D  e.  B  /\  C  e.  A  /\  ( A  i^i  B )  =  0H )  -> 
( T `  ( C  +h  D ) )  =  D )
25243com12 1195 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 968    = wceq 1374    e. wcel 1762   E.wrex 2808    i^i cin 3468    |-> cmpt 4498   ` cfv 5579   iota_crio 6235  (class class class)co 6275   ~Hchil 25498    +h cva 25499   SHcsh 25507    +H cph 25510   0Hc0h 25514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-hilex 25578  ax-hfvadd 25579  ax-hvcom 25580  ax-hvass 25581  ax-hv0cl 25582  ax-hvaddid 25583  ax-hfvmul 25584  ax-hvmulid 25585  ax-hvmulass 25586  ax-hvdistr1 25587  ax-hvdistr2 25588  ax-hvmul0 25589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-grpo 24855  df-ablo 24946  df-hvsub 25550  df-sh 25786  df-ch0 25833  df-shs 25888
This theorem is referenced by:  cdj3lem3a  27020  cdj3i  27022
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