Theorem asclfval 15004

Description: Function value of the algebra scalar lifting 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, .1.   x, .x.   x, K   x, W

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

Proof of Theorem asclfval

Dummy variables q w j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		asclfval.a	. 2 |- A = (algSc ` W )
2		df-ascl 14984	. . . . 5 |- algSc = ( w e. _V |-> ( x e. ( Base ` (Scalar ` w ) ) |-> ( x ( .s ` w ) ( 1r ` w ) ) ) )
3	2	mptrcl 5785	. . . 4 |- ( q e. (algSc ` W ) -> W e. _V )
4		mptmex 5939	. . . . 5 |- ( q e. ( x e. K |-> ( x .x. .1. ) ) -> E. j j e. K )
5		asclfval.k	. . . . . . 7 |- K = ( Base ` F )
6	5	basm 13397	. . . . . 6 |- ( j e. K -> E. k k e. F )
7	6	exlimiv 1651	. . . . 5 |- ( E. j j e. K -> E. k k e. F )
8		mptrel 4906	. . . . . . . . . . 11 |- Rel ( u e. _V |-> ( u ` (Scalar ` ndx ) ) )
9		df-slot 13339	. . . . . . . . . . . 12 |- Slot (Scalar ` ndx ) = ( u e. _V |-> ( u ` (Scalar ` ndx ) ) )
10	9	releqi 4856	. . . . . . . . . . 11 |- ( Rel Slot (Scalar ` ndx ) <-> Rel ( u e. _V |-> ( u ` (Scalar ` ndx ) ) ) )
11	8, 10	mpbir 146	. . . . . . . . . 10 |- Rel Slot (Scalar ` ndx )
12		scaid 13489	. . . . . . . . . . 11 |- Scalar = Slot (Scalar ` ndx )
13	12	releqi 4856	. . . . . . . . . 10 |- ( Rel Scalar <-> Rel Slot (Scalar ` ndx ) )
14	11, 13	mpbir 146	. . . . . . . . 9 |- Rel Scalar
15		relelfvdm 5725	. . . . . . . . 9 |- ( ( Rel Scalar /\ k e. (Scalar ` W ) ) -> W e. dom Scalar )
16	14, 15	mpan 428	. . . . . . . 8 |- ( k e. (Scalar ` W ) -> W e. dom Scalar )
17		asclfval.f	. . . . . . . 8 |- F = (Scalar ` W )
18	16, 17	eleq2s 2333	. . . . . . 7 |- ( k e. F -> W e. dom Scalar )
19	18	exlimiv 1651	. . . . . 6 |- ( E. k k e. F -> W e. dom Scalar )
20	19	elexd 2835	. . . . 5 |- ( E. k k e. F -> W e. _V )
21	4, 7, 20	3syl 17	. . . 4 |- ( q e. ( x e. K |-> ( x .x. .1. ) ) -> W e. _V )
22		fveq2 5693	. . . . . . . . . 10 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
23	22, 17	eqtr4di 2289	. . . . . . . . 9 |- ( w = W -> (Scalar ` w ) = F )
24	23	fveq2d 5697	. . . . . . . 8 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` F ) )
25	24, 5	eqtr4di 2289	. . . . . . 7 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
26		fveq2 5693	. . . . . . . . 9 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
27		asclfval.s	. . . . . . . . 9 |- .x. = ( .s ` W )
28	26, 27	eqtr4di 2289	. . . . . . . 8 |- ( w = W -> ( .s ` w ) = .x. )
29		eqidd 2239	. . . . . . . 8 |- ( w = W -> x = x )
30		fveq2 5693	. . . . . . . . 9 |- ( w = W -> ( 1r ` w ) = ( 1r ` W ) )
31		asclfval.o	. . . . . . . . 9 |- .1. = ( 1r ` W )
32	30, 31	eqtr4di 2289	. . . . . . . 8 |- ( w = W -> ( 1r ` w ) = .1. )
33	28, 29, 32	oveq123d 6100	. . . . . . 7 |- ( w = W -> ( x ( .s ` w ) ( 1r ` w ) ) = ( x .x. .1. ) )
34	25, 33	mpteq12dv 4211	. . . . . 6 |- ( w = W -> ( x e. ( Base ` (Scalar ` w ) ) |-> ( x ( .s ` w ) ( 1r ` w ) ) ) = ( x e. K |-> ( x .x. .1. ) ) )
35		id 19	. . . . . 6 |- ( W e. _V -> W e. _V )
36		basfn 13394	. . . . . . . . 9 |- Base Fn _V
37		scaslid 13490	. . . . . . . . . . 11 |- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
38	37	slotex 13362	. . . . . . . . . 10 |- ( W e. _V -> (Scalar ` W ) e. _V )
39	17, 38	eqeltrid 2325	. . . . . . . . 9 |- ( W e. _V -> F e. _V )
40		funfvex 5710	. . . . . . . . . 10 |- ( ( Fun Base /\ F e. dom Base ) -> ( Base ` F ) e. _V )
41	40	funfni 5481	. . . . . . . . 9 |- ( ( Base Fn _V /\ F e. _V ) -> ( Base ` F ) e. _V )
42	36, 39, 41	sylancr 418	. . . . . . . 8 |- ( W e. _V -> ( Base ` F ) e. _V )
43	5, 42	eqeltrid 2325	. . . . . . 7 |- ( W e. _V -> K e. _V )
44	43	mptexd 5938	. . . . . 6 |- ( W e. _V -> ( x e. K |-> ( x .x. .1. ) ) e. _V )
45	2, 34, 35, 44	fvmptd3 5796	. . . . 5 |- ( W e. _V -> (algSc ` W ) = ( x e. K |-> ( x .x. .1. ) ) )
46	45	eleq2d 2308	. . . 4 |- ( W e. _V -> ( q e. (algSc ` W ) <-> q e. ( x e. K |-> ( x .x. .1. ) ) ) )
47	3, 21, 46	pm5.21nii 716	. . 3 |- ( q e. (algSc ` W ) <-> q e. ( x e. K |-> ( x .x. .1. ) ) )
48	47	eqriv 2235	. 2 |- (algSc ` W ) = ( x e. K |-> ( x .x. .1. ) )
49	1, 48	eqtri 2259	1 |- A = ( x e. K |-> ( x .x. .1. ) )

Syntax hints:   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   |-> cmpt 4190   dom cdm 4772   Rel wrel 4777   Fn wfn 5370   ` cfv 5375  (class class class)co 6079   ndxcnx 13332  Slot cslot 13334   Basecbs 13335  Scalarcsca 13417   .scvsca 13418   1rcur 14245  algSccascl 14981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-ndx 13338  df-slot 13339  df-base 13341  df-sca 13430  df-ascl 14984

This theorem is referenced by:  asclvald  15005  asclfnd  15006  asclf  15007  rnascl  15017  ressascl  15022  asclpropd  15023