Theorem asclfval 19334

Description: Function value of the algebraic scalars function. (Contributed by Mario Carneiro, 8-Mar-2015.)

Hypotheses

Ref	Expression
asclfval.a	|- A = (algSc ` W )
asclfval.f	|- F = (Scalar ` W )
asclfval.k	|- K = ( Base ` F )
asclfval.s	|- .x. = ( .s ` W )
asclfval.o	|- .1. = ( 1r ` W )

Assertion

Ref	Expression
asclfval	|- A = ( x e. K |-> ( x .x. .1. ) )

Distinct variable groups:   x, K   x, .1.   x, .x.   x, W

Allowed substitution hints:   A( x)   F( x)

Proof of Theorem asclfval

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		asclfval.a	. 2 |- A = (algSc ` W )
2		fveq2 6191	. . . . . . . 8 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
3		asclfval.f	. . . . . . . 8 |- F = (Scalar ` W )
4	2, 3	syl6eqr 2674	. . . . . . 7 |- ( w = W -> (Scalar ` w ) = F )
5	4	fveq2d 6195	. . . . . 6 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` F ) )
6		asclfval.k	. . . . . 6 |- K = ( Base ` F )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
8		fveq2 6191	. . . . . . 7 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
9		asclfval.s	. . . . . . 7 |- .x. = ( .s ` W )
10	8, 9	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( .s ` w ) = .x. )
11		eqidd 2623	. . . . . 6 |- ( w = W -> x = x )
12		fveq2 6191	. . . . . . 7 |- ( w = W -> ( 1r ` w ) = ( 1r ` W ) )
13		asclfval.o	. . . . . . 7 |- .1. = ( 1r ` W )
14	12, 13	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( 1r ` w ) = .1. )
15	10, 11, 14	oveq123d 6671	. . . . 5 |- ( w = W -> ( x ( .s ` w ) ( 1r ` w ) ) = ( x .x. .1. ) )
16	7, 15	mpteq12dv 4733	. . . 4 |- ( w = W -> ( x e. ( Base ` (Scalar ` w ) ) |-> ( x ( .s ` w ) ( 1r ` w ) ) ) = ( x e. K |-> ( x .x. .1. ) ) )
17		df-ascl 19314	. . . 4 |- algSc = ( w e. _V |-> ( x e. ( Base ` (Scalar ` w ) ) |-> ( x ( .s ` w ) ( 1r ` w ) ) ) )
18	3	fveq2i 6194	. . . . . . 7 |- ( Base ` F ) = ( Base ` (Scalar ` W ) )
19	6, 18	eqtri 2644	. . . . . 6 |- K = ( Base ` (Scalar ` W ) )
20		fvex 6201	. . . . . 6 |- ( Base ` (Scalar ` W ) ) e. _V
21	19, 20	eqeltri 2697	. . . . 5 |- K e. _V
22	21	mptex 6486	. . . 4 |- ( x e. K |-> ( x .x. .1. ) ) e. _V
23	16, 17, 22	fvmpt 6282	. . 3 |- ( W e. _V -> (algSc ` W ) = ( x e. K |-> ( x .x. .1. ) ) )
24		fvprc 6185	. . . . 5 |- ( -. W e. _V -> (algSc ` W ) = (/) )
25		mpt0 6021	. . . . 5 |- ( x e. (/) |-> ( x .x. .1. ) ) = (/)
26	24, 25	syl6eqr 2674	. . . 4 |- ( -. W e. _V -> (algSc ` W ) = ( x e. (/) |-> ( x .x. .1. ) ) )
27		fvprc 6185	. . . . . . . . 9 |- ( -. W e. _V -> (Scalar ` W ) = (/) )
28	3, 27	syl5eq 2668	. . . . . . . 8 |- ( -. W e. _V -> F = (/) )
29	28	fveq2d 6195	. . . . . . 7 |- ( -. W e. _V -> ( Base ` F ) = ( Base ` (/) ) )
30		base0 15912	. . . . . . 7 |- (/) = ( Base ` (/) )
31	29, 30	syl6eqr 2674	. . . . . 6 |- ( -. W e. _V -> ( Base ` F ) = (/) )
32	6, 31	syl5eq 2668	. . . . 5 |- ( -. W e. _V -> K = (/) )
33	32	mpteq1d 4738	. . . 4 |- ( -. W e. _V -> ( x e. K |-> ( x .x. .1. ) ) = ( x e. (/) |-> ( x .x. .1. ) ) )
34	26, 33	eqtr4d 2659	. . 3 |- ( -. W e. _V -> (algSc ` W ) = ( x e. K |-> ( x .x. .1. ) ) )
35	23, 34	pm2.61i 176	. 2 |- (algSc ` W ) = ( x e. K |-> ( x .x. .1. ) )
36	1, 35	eqtri 2644	1 |- A = ( x e. K |-> ( x .x. .1. ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   1rcur 18501  algSccascl 19311

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

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-slot 15861  df-base 15863  df-ascl 19314

This theorem is referenced by:  asclval  19335  asclfn  19336  asclf  19337  rnascl  19343  ressascl  19344  asclpropd  19346