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Theorem cdj3lem2 27030
Description: Lemma for cdj3i 27036. Value of the first-component function  S. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdj3lem2.1  |-  A  e.  SH
cdj3lem2.2  |-  B  e.  SH
cdj3lem2.3  |-  S  =  ( x  e.  ( A  +H  B ) 
|->  ( iota_ z  e.  A  E. w  e.  B  x  =  ( z  +h  w ) ) )
Assertion
Ref Expression
cdj3lem2  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( S `  ( C  +h  D ) )  =  C )
Distinct variable groups:    x, z, w, A    x, B, z, w    x, C, z, w    x, D, z, w
Allowed substitution hints:    S( x, z, w)

Proof of Theorem cdj3lem2
StepHypRef Expression
1 cdj3lem2.1 . . . . 5  |-  A  e.  SH
2 cdj3lem2.2 . . . . 5  |-  B  e.  SH
31, 2shsvai 25958 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B )  ->  ( C  +h  D
)  e.  ( A  +H  B ) )
4 eqeq1 2471 . . . . . . 7  |-  ( x  =  ( C  +h  D )  ->  (
x  =  ( z  +h  w )  <->  ( C  +h  D )  =  ( z  +h  w ) ) )
54rexbidv 2973 . . . . . 6  |-  ( x  =  ( C  +h  D )  ->  ( E. w  e.  B  x  =  ( z  +h  w )  <->  E. w  e.  B  ( C  +h  D )  =  ( z  +h  w ) ) )
65riotabidv 6245 . . . . 5  |-  ( x  =  ( C  +h  D )  ->  ( iota_ z  e.  A  E. w  e.  B  x  =  ( z  +h  w ) )  =  ( iota_ z  e.  A  E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w ) ) )
7 cdj3lem2.3 . . . . 5  |-  S  =  ( x  e.  ( A  +H  B ) 
|->  ( iota_ z  e.  A  E. w  e.  B  x  =  ( z  +h  w ) ) )
8 riotaex 6247 . . . . 5  |-  ( iota_ z  e.  A  E. w  e.  B  ( C  +h  D )  =  ( z  +h  w ) )  e.  _V
96, 7, 8fvmpt 5948 . . . 4  |-  ( ( C  +h  D )  e.  ( A  +H  B )  ->  ( S `  ( C  +h  D ) )  =  ( iota_ z  e.  A  E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w ) ) )
103, 9syl 16 . . 3  |-  ( ( C  e.  A  /\  D  e.  B )  ->  ( S `  ( C  +h  D ) )  =  ( iota_ z  e.  A  E. w  e.  B  ( C  +h  D )  =  ( z  +h  w ) ) )
11103adant3 1016 . 2  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( S `  ( C  +h  D ) )  =  ( iota_ z  e.  A  E. w  e.  B  ( C  +h  D )  =  ( z  +h  w ) ) )
12 eqid 2467 . . . . 5  |-  ( C  +h  D )  =  ( C  +h  D
)
13 oveq2 6290 . . . . . . 7  |-  ( w  =  D  ->  ( C  +h  w )  =  ( C  +h  D
) )
1413eqeq2d 2481 . . . . . 6  |-  ( w  =  D  ->  (
( C  +h  D
)  =  ( C  +h  w )  <->  ( C  +h  D )  =  ( C  +h  D ) ) )
1514rspcev 3214 . . . . 5  |-  ( ( D  e.  B  /\  ( C  +h  D
)  =  ( C  +h  D ) )  ->  E. w  e.  B  ( C  +h  D
)  =  ( C  +h  w ) )
1612, 15mpan2 671 . . . 4  |-  ( D  e.  B  ->  E. w  e.  B  ( C  +h  D )  =  ( C  +h  w ) )
17163ad2ant2 1018 . . 3  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  E. w  e.  B  ( C  +h  D
)  =  ( C  +h  w ) )
18 simp1 996 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  C  e.  A )
191, 2cdjreui 27027 . . . . . 6  |-  ( ( ( C  +h  D
)  e.  ( A  +H  B )  /\  ( A  i^i  B )  =  0H )  ->  E! z  e.  A  E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w ) )
203, 19sylan 471 . . . . 5  |-  ( ( ( C  e.  A  /\  D  e.  B
)  /\  ( A  i^i  B )  =  0H )  ->  E! z  e.  A  E. w  e.  B  ( C  +h  D )  =  ( z  +h  w ) )
21203impa 1191 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  E! z  e.  A  E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w ) )
22 oveq1 6289 . . . . . . 7  |-  ( z  =  C  ->  (
z  +h  w )  =  ( C  +h  w ) )
2322eqeq2d 2481 . . . . . 6  |-  ( z  =  C  ->  (
( C  +h  D
)  =  ( z  +h  w )  <->  ( C  +h  D )  =  ( C  +h  w ) ) )
2423rexbidv 2973 . . . . 5  |-  ( z  =  C  ->  ( E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w )  <->  E. w  e.  B  ( C  +h  D )  =  ( C  +h  w ) ) )
2524riota2 6266 . . . 4  |-  ( ( C  e.  A  /\  E! z  e.  A  E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w ) )  ->  ( E. w  e.  B  ( C  +h  D )  =  ( C  +h  w )  <-> 
( iota_ z  e.  A  E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w ) )  =  C ) )
2618, 21, 25syl2anc 661 . . 3  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( E. w  e.  B  ( C  +h  D )  =  ( C  +h  w )  <-> 
( iota_ z  e.  A  E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w ) )  =  C ) )
2717, 26mpbid 210 . 2  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( iota_ z  e.  A  E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w ) )  =  C )
2811, 27eqtrd 2508 1  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( S `  ( C  +h  D ) )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   E!wreu 2816    i^i cin 3475    |-> cmpt 4505   ` cfv 5586   iota_crio 6242  (class class class)co 6282    +h cva 25513   SHcsh 25521    +H cph 25524   0Hc0h 25528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-hilex 25592  ax-hfvadd 25593  ax-hvcom 25594  ax-hvass 25595  ax-hv0cl 25596  ax-hvaddid 25597  ax-hfvmul 25598  ax-hvmulid 25599  ax-hvmulass 25600  ax-hvdistr1 25601  ax-hvdistr2 25602  ax-hvmul0 25603
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-grpo 24869  df-ablo 24960  df-hvsub 25564  df-sh 25800  df-ch0 25847  df-shs 25902
This theorem is referenced by:  cdj3lem2a  27031  cdj3lem2b  27032  cdj3lem3  27033  cdj3i  27036
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