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Theorem cdj3lem2 28088
Description: Lemma for cdj3i 28094. 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 27017 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B )  ->  ( C  +h  D
)  e.  ( A  +H  B ) )
4 eqeq1 2455 . . . . . . 7  |-  ( x  =  ( C  +h  D )  ->  (
x  =  ( z  +h  w )  <->  ( C  +h  D )  =  ( z  +h  w ) ) )
54rexbidv 2901 . . . . . 6  |-  ( x  =  ( C  +h  D )  ->  ( E. w  e.  B  x  =  ( z  +h  w )  <->  E. w  e.  B  ( C  +h  D )  =  ( z  +h  w ) ) )
65riotabidv 6254 . . . . 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 6256 . . . . 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 17 . . 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 1028 . 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 2451 . . . . 5  |-  ( C  +h  D )  =  ( C  +h  D
)
13 oveq2 6298 . . . . . . 7  |-  ( w  =  D  ->  ( C  +h  w )  =  ( C  +h  D
) )
1413eqeq2d 2461 . . . . . 6  |-  ( w  =  D  ->  (
( C  +h  D
)  =  ( C  +h  w )  <->  ( C  +h  D )  =  ( C  +h  D ) ) )
1514rspcev 3150 . . . . 5  |-  ( ( D  e.  B  /\  ( C  +h  D
)  =  ( C  +h  D ) )  ->  E. w  e.  B  ( C  +h  D
)  =  ( C  +h  w ) )
1612, 15mpan2 677 . . . 4  |-  ( D  e.  B  ->  E. w  e.  B  ( C  +h  D )  =  ( C  +h  w ) )
17163ad2ant2 1030 . . 3  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  E. w  e.  B  ( C  +h  D
)  =  ( C  +h  w ) )
18 simp1 1008 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  C  e.  A )
191, 2cdjreui 28085 . . . . 5  |-  ( ( ( 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, 19stoic3 1660 . . . 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 ) )
21 oveq1 6297 . . . . . . 7  |-  ( z  =  C  ->  (
z  +h  w )  =  ( C  +h  w ) )
2221eqeq2d 2461 . . . . . 6  |-  ( z  =  C  ->  (
( C  +h  D
)  =  ( z  +h  w )  <->  ( C  +h  D )  =  ( C  +h  w ) ) )
2322rexbidv 2901 . . . . 5  |-  ( z  =  C  ->  ( E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w )  <->  E. w  e.  B  ( C  +h  D )  =  ( C  +h  w ) ) )
2423riota2 6274 . . . 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 ) )
2518, 20, 24syl2anc 667 . . 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 ) )
2617, 25mpbid 214 . 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 )
2711, 26eqtrd 2485 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   E.wrex 2738   E!wreu 2739    i^i cin 3403    |-> cmpt 4461   ` cfv 5582   iota_crio 6251  (class class class)co 6290    +h cva 26573   SHcsh 26581    +H cph 26584   0Hc0h 26588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-hilex 26652  ax-hfvadd 26653  ax-hvcom 26654  ax-hvass 26655  ax-hv0cl 26656  ax-hvaddid 26657  ax-hfvmul 26658  ax-hvmulid 26659  ax-hvmulass 26660  ax-hvdistr1 26661  ax-hvdistr2 26662  ax-hvmul0 26663
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-grpo 25919  df-ablo 26010  df-hvsub 26624  df-sh 26860  df-ch0 26906  df-shs 26961
This theorem is referenced by:  cdj3lem2a  28089  cdj3lem2b  28090  cdj3lem3  28091  cdj3i  28094
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