Theorem smfval 27460

Description: Value of the function for the scalar multiplication operation on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)

Hypothesis

Ref	Expression
smfval.4	|- S = ( .sOLD ` U )

Assertion

Ref	Expression
smfval	|- S = ( 2nd ` ( 1st ` U ) )

Proof of Theorem smfval

Step	Hyp	Ref	Expression
1		smfval.4	. 2 |- S = ( .sOLD ` U )
2		df-sm 27452	. . . . 5 |- .sOLD = ( 2nd o. 1st )
3	2	fveq1i 6192	. . . 4 |- ( .sOLD ` U ) = ( ( 2nd o. 1st ) ` U )
4		fo1st 7188	. . . . . 6 |- 1st : _V -onto-> _V
5		fof 6115	. . . . . 6 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
6	4, 5	ax-mp 5	. . . . 5 |- 1st : _V --> _V
7		fvco3 6275	. . . . 5 |- ( ( 1st : _V --> _V /\ U e. _V ) -> ( ( 2nd o. 1st ) ` U ) = ( 2nd ` ( 1st ` U ) ) )
8	6, 7	mpan 706	. . . 4 |- ( U e. _V -> ( ( 2nd o. 1st ) ` U ) = ( 2nd ` ( 1st ` U ) ) )
9	3, 8	syl5eq 2668	. . 3 |- ( U e. _V -> ( .sOLD ` U ) = ( 2nd ` ( 1st ` U ) ) )
10		fvprc 6185	. . . 4 |- ( -. U e. _V -> ( .sOLD ` U ) = (/) )
11		fvprc 6185	. . . . . 6 |- ( -. U e. _V -> ( 1st ` U ) = (/) )
12	11	fveq2d 6195	. . . . 5 |- ( -. U e. _V -> ( 2nd ` ( 1st ` U ) ) = ( 2nd ` (/) ) )
13		2nd0 7175	. . . . 5 |- ( 2nd ` (/) ) = (/)
14	12, 13	syl6req 2673	. . . 4 |- ( -. U e. _V -> (/) = ( 2nd ` ( 1st ` U ) ) )
15	10, 14	eqtrd 2656	. . 3 |- ( -. U e. _V -> ( .sOLD ` U ) = ( 2nd ` ( 1st ` U ) ) )
16	9, 15	pm2.61i 176	. 2 |- ( .sOLD ` U ) = ( 2nd ` ( 1st ` U ) )
17	1, 16	eqtri 2644	1 |- S = ( 2nd ` ( 1st ` U ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   o. ccom 5118   -->wf 5884   -onto->wfo 5886   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   .sOLDcns 27442

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-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-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-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-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-sm 27452

This theorem is referenced by:  nvvop  27464  nvsf  27474  nvscl  27481  nvsid  27482  nvsass  27483  nvdi  27485  nvdir  27486  nv2  27487  nv0  27492  nvsz  27493  nvinv  27494  nvtri  27525  cnnvs  27535  phop  27673  phpar  27679  ipdirilem  27684  h2hsm  27832  hhsssm  28115