Theorem suppssof1 7328

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.

Step	Hyp	Ref	Expression
1		suppssof1.a	. . . . 5 |- ( ph -> A : D --> V )
2		ffn 6045	. . . . 5 |- ( A : D --> V -> A Fn D )
3	1, 2	syl 17	. . . 4 |- ( ph -> A Fn D )
4		suppssof1.b	. . . . 5 |- ( ph -> B : D --> R )
5		ffn 6045	. . . . 5 |- ( B : D --> R -> B Fn D )
6	4, 5	syl 17	. . . 4 |- ( ph -> B Fn D )
7		suppssof1.d	. . . 4 |- ( ph -> D e. W )
8		inidm 3822	. . . 4 |- ( D i^i D ) = D
9		eqidd 2623	. . . 4 |- ( ( ph /\ x e. D ) -> ( A ` x ) = ( A ` x ) )
10		eqidd 2623	. . . 4 |- ( ( ph /\ x e. D ) -> ( B ` x ) = ( B ` x ) )
11	3, 6, 7, 7, 8, 9, 10	offval 6904	. . 3 |- ( ph -> ( A oF O B ) = ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) )
12	11	oveq1d 6665	. 2 |- ( ph -> ( ( A oF O B ) supp Z ) = ( ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) supp Z ) )
13	1	feqmptd 6249	. . . . 5 |- ( ph -> A = ( x e. D |-> ( A ` x ) ) )
14	13	oveq1d 6665	. . . 4 |- ( ph -> ( A supp Y ) = ( ( x e. D |-> ( A ` x ) ) supp Y ) )
15		suppssof1.s	. . . 4 |- ( ph -> ( A supp Y ) C_ L )
16	14, 15	eqsstr3d 3640	. . 3 |- ( ph -> ( ( x e. D |-> ( A ` x ) ) supp Y ) C_ L )
17		suppssof1.o	. . 3 |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z )
18		fvexd 6203	. . 3 |- ( ( ph /\ x e. D ) -> ( A ` x ) e. _V )
19	4	ffvelrnda 6359	. . 3 |- ( ( ph /\ x e. D ) -> ( B ` x ) e. R )
20		suppssof1.y	. . 3 |- ( ph -> Y e. U )
21	16, 17, 18, 19, 20	suppssov1 7327	. 2 |- ( ph -> ( ( x e. D |-> ( ( A ` x ) O ( B ` x ) ) ) supp Z ) C_ L )
22	12, 21	eqsstrd 3639	1 |- ( ph -> ( ( A oF O B ) supp Z ) C_ L )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   supp csupp 7295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-supp 7296

This theorem is referenced by:  psrbagev1  19510  frlmup1  20137  jensen  24715