HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  cdj3lem2 Unicode version

Theorem cdj3lem2 23031
Description: Lemma for cdj3i 23037. 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 21959 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B )  ->  ( C  +h  D
)  e.  ( A  +H  B ) )
4 eqeq1 2302 . . . . . . 7  |-  ( x  =  ( C  +h  D )  ->  (
x  =  ( z  +h  w )  <->  ( C  +h  D )  =  ( z  +h  w ) ) )
54rexbidv 2577 . . . . . 6  |-  ( x  =  ( C  +h  D )  ->  ( E. w  e.  B  x  =  ( z  +h  w )  <->  E. w  e.  B  ( C  +h  D )  =  ( z  +h  w ) ) )
65riotabidv 6322 . . . . 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 6324 . . . . 5  |-  ( iota_ z  e.  A E. w  e.  B  ( C  +h  D )  =  ( z  +h  w ) )  e.  _V
96, 7, 8fvmpt 5618 . . . 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 15 . . 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 975 . 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 2296 . . . . 5  |-  ( C  +h  D )  =  ( C  +h  D
)
13 oveq2 5882 . . . . . . 7  |-  ( w  =  D  ->  ( C  +h  w )  =  ( C  +h  D
) )
1413eqeq2d 2307 . . . . . 6  |-  ( w  =  D  ->  (
( C  +h  D
)  =  ( C  +h  w )  <->  ( C  +h  D )  =  ( C  +h  D ) ) )
1514rspcev 2897 . . . . 5  |-  ( ( D  e.  B  /\  ( C  +h  D
)  =  ( C  +h  D ) )  ->  E. w  e.  B  ( C  +h  D
)  =  ( C  +h  w ) )
1612, 15mpan2 652 . . . 4  |-  ( D  e.  B  ->  E. w  e.  B  ( C  +h  D )  =  ( C  +h  w ) )
17163ad2ant2 977 . . 3  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  E. w  e.  B  ( C  +h  D
)  =  ( C  +h  w ) )
18 simp1 955 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  C  e.  A )
191, 2cdjreui 23028 . . . . . 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 457 . . . . 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 1146 . . . 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 5881 . . . . . . 7  |-  ( z  =  C  ->  (
z  +h  w )  =  ( C  +h  w ) )
2322eqeq2d 2307 . . . . . 6  |-  ( z  =  C  ->  (
( C  +h  D
)  =  ( z  +h  w )  <->  ( C  +h  D )  =  ( C  +h  w ) ) )
2423rexbidv 2577 . . . . 5  |-  ( z  =  C  ->  ( E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w )  <->  E. w  e.  B  ( C  +h  D )  =  ( C  +h  w ) ) )
2524riota2 6343 . . . 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 642 . . 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 201 . 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 2328 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   E!wreu 2558    i^i cin 3164    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   iota_crio 6313    +h cva 21516   SHcsh 21524    +H cph 21527   0Hc0h 21531
This theorem is referenced by:  cdj3lem2a  23032  cdj3lem2b  23033  cdj3lem3  23034  cdj3i  23037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-grpo 20874  df-ablo 20965  df-hvsub 21567  df-sh 21802  df-ch0 21848  df-shs 21903
  Copyright terms: Public domain W3C validator