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Theorem cdj3lem2 27753
Description: Lemma for cdj3i 27759. 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 26682 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B )  ->  ( C  +h  D
)  e.  ( A  +H  B ) )
4 eqeq1 2406 . . . . . . 7  |-  ( x  =  ( C  +h  D )  ->  (
x  =  ( z  +h  w )  <->  ( C  +h  D )  =  ( z  +h  w ) ) )
54rexbidv 2917 . . . . . 6  |-  ( x  =  ( C  +h  D )  ->  ( E. w  e.  B  x  =  ( z  +h  w )  <->  E. w  e.  B  ( C  +h  D )  =  ( z  +h  w ) ) )
65riotabidv 6241 . . . . 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 6243 . . . . 5  |-  ( iota_ z  e.  A  E. w  e.  B  ( C  +h  D )  =  ( z  +h  w ) )  e.  _V
96, 7, 8fvmpt 5931 . . . 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 1017 . 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 2402 . . . . 5  |-  ( C  +h  D )  =  ( C  +h  D
)
13 oveq2 6285 . . . . . . 7  |-  ( w  =  D  ->  ( C  +h  w )  =  ( C  +h  D
) )
1413eqeq2d 2416 . . . . . 6  |-  ( w  =  D  ->  (
( C  +h  D
)  =  ( C  +h  w )  <->  ( C  +h  D )  =  ( C  +h  D ) ) )
1514rspcev 3159 . . . . 5  |-  ( ( D  e.  B  /\  ( C  +h  D
)  =  ( C  +h  D ) )  ->  E. w  e.  B  ( C  +h  D
)  =  ( C  +h  w ) )
1612, 15mpan2 669 . . . 4  |-  ( D  e.  B  ->  E. w  e.  B  ( C  +h  D )  =  ( C  +h  w ) )
17163ad2ant2 1019 . . 3  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  E. w  e.  B  ( C  +h  D
)  =  ( C  +h  w ) )
18 simp1 997 . . . 4  |-  ( ( C  e.  A  /\  D  e.  B  /\  ( A  i^i  B )  =  0H )  ->  C  e.  A )
191, 2cdjreui 27750 . . . . 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 1630 . . . 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 6284 . . . . . . 7  |-  ( z  =  C  ->  (
z  +h  w )  =  ( C  +h  w ) )
2221eqeq2d 2416 . . . . . 6  |-  ( z  =  C  ->  (
( C  +h  D
)  =  ( z  +h  w )  <->  ( C  +h  D )  =  ( C  +h  w ) ) )
2322rexbidv 2917 . . . . 5  |-  ( z  =  C  ->  ( E. w  e.  B  ( C  +h  D
)  =  ( z  +h  w )  <->  E. w  e.  B  ( C  +h  D )  =  ( C  +h  w ) ) )
2423riota2 6261 . . . 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 659 . . 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 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 )
2711, 26eqtrd 2443 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   E.wrex 2754   E!wreu 2755    i^i cin 3412    |-> cmpt 4452   ` cfv 5568   iota_crio 6238  (class class class)co 6277    +h cva 26237   SHcsh 26245    +H cph 26248   0Hc0h 26252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-hilex 26316  ax-hfvadd 26317  ax-hvcom 26318  ax-hvass 26319  ax-hv0cl 26320  ax-hvaddid 26321  ax-hfvmul 26322  ax-hvmulid 26323  ax-hvmulass 26324  ax-hvdistr1 26325  ax-hvdistr2 26326  ax-hvmul0 26327
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-grpo 25593  df-ablo 25684  df-hvsub 26288  df-sh 26524  df-ch0 26571  df-shs 26626
This theorem is referenced by:  cdj3lem2a  27754  cdj3lem2b  27755  cdj3lem3  27756  cdj3i  27759
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