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Theorem suppssof1 6873
Description: Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
Hypotheses
Ref Expression
suppssof1.s  |-  ( ph  ->  ( A supp  Y ) 
C_  L )
suppssof1.o  |-  ( (
ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )
suppssof1.a  |-  ( ph  ->  A : D --> V )
suppssof1.b  |-  ( ph  ->  B : D --> R )
suppssof1.d  |-  ( ph  ->  D  e.  W )
suppssof1.y  |-  ( ph  ->  Y  e.  U )
Assertion
Ref Expression
suppssof1  |-  ( ph  ->  ( ( A  oF O B ) supp 
Z )  C_  L
)
Distinct variable groups:    ph, v    v, B    v, O    v, R    v, Y    v, Z
Allowed substitution hints:    A( v)    D( v)    U( v)    L( v)    V( v)    W( v)

Proof of Theorem suppssof1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 suppssof1.a . . . . 5  |-  ( ph  ->  A : D --> V )
2 ffn 5656 . . . . 5  |-  ( A : D --> V  ->  A  Fn  D )
31, 2syl 16 . . . 4  |-  ( ph  ->  A  Fn  D )
4 suppssof1.b . . . . 5  |-  ( ph  ->  B : D --> R )
5 ffn 5656 . . . . 5  |-  ( B : D --> R  ->  B  Fn  D )
64, 5syl 16 . . . 4  |-  ( ph  ->  B  Fn  D )
7 suppssof1.d . . . 4  |-  ( ph  ->  D  e.  W )
8 inidm 3638 . . . 4  |-  ( D  i^i  D )  =  D
9 eqidd 2397 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  =  ( A `  x ) )
10 eqidd 2397 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  =  ( B `  x ) )
113, 6, 7, 7, 8, 9, 10offval 6468 . . 3  |-  ( ph  ->  ( A  oF O B )  =  ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) ) )
1211oveq1d 6233 . 2  |-  ( ph  ->  ( ( A  oF O B ) supp 
Z )  =  ( ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) ) supp  Z ) )
131feqmptd 5844 . . . . 5  |-  ( ph  ->  A  =  ( x  e.  D  |->  ( A `
 x ) ) )
1413oveq1d 6233 . . . 4  |-  ( ph  ->  ( A supp  Y )  =  ( ( x  e.  D  |->  ( A `
 x ) ) supp 
Y ) )
15 suppssof1.s . . . 4  |-  ( ph  ->  ( A supp  Y ) 
C_  L )
1614, 15eqsstr3d 3469 . . 3  |-  ( ph  ->  ( ( x  e.  D  |->  ( A `  x ) ) supp  Y
)  C_  L )
17 suppssof1.o . . 3  |-  ( (
ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )
18 fvex 5801 . . . 4  |-  ( A `
 x )  e. 
_V
1918a1i 11 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  e.  _V )
204ffvelrnda 5950 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  e.  R )
21 suppssof1.y . . 3  |-  ( ph  ->  Y  e.  U )
2216, 17, 19, 20, 21suppssov1 6872 . 2  |-  ( ph  ->  ( ( x  e.  D  |->  ( ( A `
 x ) O ( B `  x
) ) ) supp  Z
)  C_  L )
2312, 22eqsstrd 3468 1  |-  ( ph  ->  ( ( A  oF O B ) supp 
Z )  C_  L
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   _Vcvv 3051    C_ wss 3406    |-> cmpt 4442    Fn wfn 5508   -->wf 5509   ` cfv 5513  (class class class)co 6218    oFcof 6459   supp csupp 6839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-of 6461  df-supp 6840
This theorem is referenced by:  psrbagev1  18313  frlmup1  18941  jensen  23458
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