Theorem resvval 29827

Description: Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)

Hypotheses

Ref	Expression
resvsca.r	|- R = ( W ↾v A )
resvsca.f	|- F = (Scalar ` W )
resvsca.b	|- B = ( Base ` F )

Assertion

Ref	Expression
resvval	|- ( ( W e. X /\ A e. Y ) -> R = if ( B C_ A , W , ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) ) )

Proof of Theorem resvval

Dummy variables x w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		resvsca.r	. 2 |- R = ( W ↾v A )
2		elex 3212	. . 3 |- ( W e. X -> W e. _V )
3		elex 3212	. . 3 |- ( A e. Y -> A e. _V )
4		ovex 6678	. . . . . 6 |- ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) e. _V
5		ifcl 4130	. . . . . 6 |- ( ( W e. _V /\ ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) e. _V ) -> if ( B C_ A , W , ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) ) e. _V )
6	4, 5	mpan2 707	. . . . 5 |- ( W e. _V -> if ( B C_ A , W , ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) ) e. _V )
7	6	adantr 481	. . . 4 |- ( ( W e. _V /\ A e. _V ) -> if ( B C_ A , W , ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) ) e. _V )
8		simpl 473	. . . . . . . . . . 11 |- ( ( w = W /\ x = A ) -> w = W )
9	8	fveq2d 6195	. . . . . . . . . 10 |- ( ( w = W /\ x = A ) -> (Scalar ` w ) = (Scalar ` W ) )
10		resvsca.f	. . . . . . . . . 10 |- F = (Scalar ` W )
11	9, 10	syl6eqr 2674	. . . . . . . . 9 |- ( ( w = W /\ x = A ) -> (Scalar ` w ) = F )
12	11	fveq2d 6195	. . . . . . . 8 |- ( ( w = W /\ x = A ) -> ( Base ` (Scalar ` w ) ) = ( Base ` F ) )
13		resvsca.b	. . . . . . . 8 |- B = ( Base ` F )
14	12, 13	syl6eqr 2674	. . . . . . 7 |- ( ( w = W /\ x = A ) -> ( Base ` (Scalar ` w ) ) = B )
15		simpr 477	. . . . . . 7 |- ( ( w = W /\ x = A ) -> x = A )
16	14, 15	sseq12d 3634	. . . . . 6 |- ( ( w = W /\ x = A ) -> ( ( Base ` (Scalar ` w ) ) C_ x <-> B C_ A ) )
17	11, 15	oveq12d 6668	. . . . . . . 8 |- ( ( w = W /\ x = A ) -> ( (Scalar ` w ) ↾s x ) = ( F ↾s A ) )
18	17	opeq2d 4409	. . . . . . 7 |- ( ( w = W /\ x = A ) -> <. (Scalar ` ndx ) , ( (Scalar ` w ) ↾s x ) >. = <. (Scalar ` ndx ) , ( F ↾s A ) >. )
19	8, 18	oveq12d 6668	. . . . . 6 |- ( ( w = W /\ x = A ) -> ( w sSet <. (Scalar ` ndx ) , ( (Scalar ` w ) ↾s x ) >. ) = ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) )
20	16, 8, 19	ifbieq12d 4113	. . . . 5 |- ( ( w = W /\ x = A ) -> if ( ( Base ` (Scalar ` w ) ) C_ x , w , ( w sSet <. (Scalar ` ndx ) , ( (Scalar ` w ) ↾s x ) >. ) ) = if ( B C_ A , W , ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) ) )
21		df-resv 29825	. . . . 5 |- ↾v = ( w e. _V , x e. _V |-> if ( ( Base ` (Scalar ` w ) ) C_ x , w , ( w sSet <. (Scalar ` ndx ) , ( (Scalar ` w ) ↾s x ) >. ) ) )
22	20, 21	ovmpt2ga 6790	. . . 4 |- ( ( W e. _V /\ A e. _V /\ if ( B C_ A , W , ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) ) e. _V ) -> ( W ↾v A ) = if ( B C_ A , W , ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) ) )
23	7, 22	mpd3an3 1425	. . 3 |- ( ( W e. _V /\ A e. _V ) -> ( W ↾v A ) = if ( B C_ A , W , ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) ) )
24	2, 3, 23	syl2an 494	. 2 |- ( ( W e. X /\ A e. Y ) -> ( W ↾v A ) = if ( B C_ A , W , ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) ) )
25	1, 24	syl5eq 2668	1 |- ( ( W e. X /\ A e. Y ) -> R = if ( B C_ A , W , ( W sSet <. (Scalar ` ndx ) , ( F ↾s A ) >. ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ifcif 4086   <.cop 4183   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855   Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944   ↾_v cresv 29824

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-resv 29825

This theorem is referenced by:  resvid2  29828  resvval2  29829