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Theorem cdj3lem2 23779
Description: Lemma for cdj3i 23785. 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 22707 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B )  ->  ( C  +h  D
)  e.  ( A  +H  B ) )
4 eqeq1 2386 . . . . . . 7  |-  ( x  =  ( C  +h  D )  ->  (
x  =  ( z  +h  w )  <->  ( C  +h  D )  =  ( z  +h  w ) ) )
54rexbidv 2663 . . . . . 6  |-  ( x  =  ( C  +h  D )  ->  ( E. w  e.  B  x  =  ( z  +h  w )  <->  E. w  e.  B  ( C  +h  D )  =  ( z  +h  w ) ) )
65riotabidv 6480 . . . . 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 6482 . . . . 5  |-  ( iota_ z  e.  A E. w  e.  B  ( C  +h  D )  =  ( z  +h  w ) )  e.  _V
96, 7, 8fvmpt 5738 . . . 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 977 . 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 2380 . . . . 5  |-  ( C  +h  D )  =  ( C  +h  D
)
13 oveq2 6021 . . . . . . 7  |-  ( w  =  D  ->  ( C  +h  w )  =  ( C  +h  D
) )
1413eqeq2d 2391 . . . . . 6  |-  ( w  =  D  ->  (
( C  +h  D
)  =  ( C  +h  w )  <->  ( C  +h  D )  =  ( C  +h  D ) ) )
1514rspcev 2988 . . . . 5  |-  ( ( D  e.  B  /\  ( C  +h  D
)  =  ( C  +h  D ) )  ->  E. w  e.  B  ( C  +h  D
)  =  ( C  +h  w ) )
1612, 15mpan2 653 . . . 4  |-  ( D  e.  B  ->  E. w  e.  B  ( C  +h  D )  =  ( C  +h  w ) )
17163ad2ant2 979 . . 3  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  E. w  e.  B  ( C  +h  D
)  =  ( C  +h  w ) )
18 simp1 957 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  C  e.  A )
191, 2cdjreui 23776 . . . . . 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 458 . . . . 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 1148 . . . 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 6020 . . . . . . 7  |-  ( z  =  C  ->  (
z  +h  w )  =  ( C  +h  w ) )
2322eqeq2d 2391 . . . . . 6  |-  ( z  =  C  ->  (
( C  +h  D
)  =  ( z  +h  w )  <->  ( C  +h  D )  =  ( C  +h  w ) ) )
2423rexbidv 2663 . . . . 5  |-  ( z  =  C  ->  ( E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w )  <->  E. w  e.  B  ( C  +h  D )  =  ( C  +h  w ) ) )
2524riota2 6501 . . . 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 643 . . 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 202 . 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 2412 1  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  -> 
( S `  ( C  +h  D ) )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2643   E!wreu 2644    i^i cin 3255    e. cmpt 4200   ` cfv 5387  (class class class)co 6013   iota_crio 6471    +h cva 22264   SHcsh 22272    +H cph 22275   0Hc0h 22279
This theorem is referenced by:  cdj3lem2a  23780  cdj3lem2b  23781  cdj3lem3  23782  cdj3i  23785
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-hilex 22343  ax-hfvadd 22344  ax-hvcom 22345  ax-hvass 22346  ax-hv0cl 22347  ax-hvaddid 22348  ax-hfvmul 22349  ax-hvmulid 22350  ax-hvmulass 22351  ax-hvdistr1 22352  ax-hvdistr2 22353  ax-hvmul0 22354
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-grpo 21620  df-ablo 21711  df-hvsub 22315  df-sh 22550  df-ch0 22596  df-shs 22651
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