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Theorem cdj3lem3a 23791
Description: Lemma for cdj3i 23793. Closure of the second-component function  T. (Contributed by NM, 26-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
cdj3lem3a  |-  ( ( C  e.  ( A  +H  B )  /\  ( A  i^i  B )  =  0H )  -> 
( T `  C
)  e.  B )
Distinct variable groups:    x, z, w, A    x, B, z, w    x, C, z, w
Allowed substitution hints:    T( x, z, w)

Proof of Theorem cdj3lem3a
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdj3lem2.1 . . . 4  |-  A  e.  SH
2 cdj3lem2.2 . . . 4  |-  B  e.  SH
31, 2shseli 22667 . . 3  |-  ( C  e.  ( A  +H  B )  <->  E. v  e.  A  E. u  e.  B  C  =  ( v  +h  u
) )
4 cdj3lem3.3 . . . . . . . . . 10  |-  T  =  ( x  e.  ( A  +H  B ) 
|->  ( iota_ w  e.  B E. z  e.  A  x  =  ( z  +h  w ) ) )
51, 2, 4cdj3lem3 23790 . . . . . . . . 9  |-  ( ( v  e.  A  /\  u  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( T `  (
v  +h  u ) )  =  u )
6 simp2 958 . . . . . . . . 9  |-  ( ( v  e.  A  /\  u  e.  B  /\  ( A  i^i  B )  =  0H )  ->  u  e.  B )
75, 6eqeltrd 2462 . . . . . . . 8  |-  ( ( v  e.  A  /\  u  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( T `  (
v  +h  u ) )  e.  B )
873expa 1153 . . . . . . 7  |-  ( ( ( v  e.  A  /\  u  e.  B
)  /\  ( A  i^i  B )  =  0H )  ->  ( T `  ( v  +h  u
) )  e.  B
)
9 fveq2 5669 . . . . . . . 8  |-  ( C  =  ( v  +h  u )  ->  ( T `  C )  =  ( T `  ( v  +h  u
) ) )
109eleq1d 2454 . . . . . . 7  |-  ( C  =  ( v  +h  u )  ->  (
( T `  C
)  e.  B  <->  ( T `  ( v  +h  u
) )  e.  B
) )
118, 10syl5ibr 213 . . . . . 6  |-  ( C  =  ( v  +h  u )  ->  (
( ( v  e.  A  /\  u  e.  B )  /\  ( A  i^i  B )  =  0H )  ->  ( T `  C )  e.  B ) )
1211exp3a 426 . . . . 5  |-  ( C  =  ( v  +h  u )  ->  (
( v  e.  A  /\  u  e.  B
)  ->  ( ( A  i^i  B )  =  0H  ->  ( T `  C )  e.  B
) ) )
1312com13 76 . . . 4  |-  ( ( A  i^i  B )  =  0H  ->  (
( v  e.  A  /\  u  e.  B
)  ->  ( C  =  ( v  +h  u )  ->  ( T `  C )  e.  B ) ) )
1413rexlimdvv 2780 . . 3  |-  ( ( A  i^i  B )  =  0H  ->  ( E. v  e.  A  E. u  e.  B  C  =  ( v  +h  u )  ->  ( T `  C )  e.  B ) )
153, 14syl5bi 209 . 2  |-  ( ( A  i^i  B )  =  0H  ->  ( C  e.  ( A  +H  B )  ->  ( T `  C )  e.  B ) )
1615impcom 420 1  |-  ( ( C  e.  ( A  +H  B )  /\  ( A  i^i  B )  =  0H )  -> 
( T `  C
)  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2651    i^i cin 3263    e. cmpt 4208   ` cfv 5395  (class class class)co 6021   iota_crio 6479    +h cva 22272   SHcsh 22280    +H cph 22283   0Hc0h 22287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-hilex 22351  ax-hfvadd 22352  ax-hvcom 22353  ax-hvass 22354  ax-hv0cl 22355  ax-hvaddid 22356  ax-hfvmul 22357  ax-hvmulid 22358  ax-hvmulass 22359  ax-hvdistr1 22360  ax-hvdistr2 22361  ax-hvmul0 22362
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-grpo 21628  df-ablo 21719  df-hvsub 22323  df-sh 22558  df-ch0 22604  df-shs 22659
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