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Theorem suppssof1 6833
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 5668 . . . . 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 5668 . . . . 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 3668 . . . 4  |-  ( D  i^i  D )  =  D
9 eqidd 2455 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  =  ( A `  x ) )
10 eqidd 2455 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  =  ( B `  x ) )
113, 6, 7, 7, 8, 9, 10offval 6438 . . 3  |-  ( ph  ->  ( A  oF O B )  =  ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) ) )
1211oveq1d 6216 . 2  |-  ( ph  ->  ( ( A  oF O B ) supp 
Z )  =  ( ( x  e.  D  |->  ( ( A `  x ) O ( B `  x ) ) ) supp  Z ) )
131feqmptd 5854 . . . . 5  |-  ( ph  ->  A  =  ( x  e.  D  |->  ( A `
 x ) ) )
1413oveq1d 6216 . . . 4  |-  ( ph  ->  ( A supp  Y )  =  ( ( x  e.  D  |->  ( A `
 x ) ) supp 
Y ) )
15 suppssof1.s . . . 4  |-  ( ph  ->  ( A supp  Y ) 
C_  L )
1614, 15eqsstr3d 3500 . . 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 5810 . . . 4  |-  ( A `
 x )  e. 
_V
1918a1i 11 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( A `  x )  e.  _V )
204ffvelrnda 5953 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( B `  x )  e.  R )
21 suppssof1.y . . 3  |-  ( ph  ->  Y  e.  U )
2216, 17, 19, 20, 21suppssov1 6832 . 2  |-  ( ph  ->  ( ( x  e.  D  |->  ( ( A `
 x ) O ( B `  x
) ) ) supp  Z
)  C_  L )
2312, 22eqsstrd 3499 1  |-  ( ph  ->  ( ( A  oF O B ) supp 
Z )  C_  L
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3437    |-> cmpt 4459    Fn wfn 5522   -->wf 5523   ` cfv 5527  (class class class)co 6201    oFcof 6429   supp csupp 6801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-supp 6802
This theorem is referenced by:  psrbagev1  17719  frlmup1  18352  jensen  22516
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