Theorem lnoadd 27613

Description: Addition property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnoadd.1	|- X = ( BaseSet ` U )
lnoadd.5	|- G = ( +v ` U )
lnoadd.6	|- H = ( +v ` W )
lnoadd.7	|- L = ( U LnOp W )

Assertion

Ref	Expression
lnoadd	|- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( A e. X /\ B e. X ) ) -> ( T ` ( A G B ) ) = ( ( T ` A ) H ( T ` B ) ) )

Proof of Theorem lnoadd

Step	Hyp	Ref	Expression
1		ax-1cn 9994	. . 3 |- 1 e. CC
2		lnoadd.1	. . . 4 |- X = ( BaseSet ` U )
3		eqid 2622	. . . 4 |- ( BaseSet ` W ) = ( BaseSet ` W )
4		lnoadd.5	. . . 4 |- G = ( +v ` U )
5		lnoadd.6	. . . 4 |- H = ( +v ` W )
6		eqid 2622	. . . 4 |- ( .sOLD ` U ) = ( .sOLD ` U )
7		eqid 2622	. . . 4 |- ( .sOLD ` W ) = ( .sOLD ` W )
8		lnoadd.7	. . . 4 |- L = ( U LnOp W )
9	2, 3, 4, 5, 6, 7, 8	lnolin 27609	. . 3 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( 1 e. CC /\ A e. X /\ B e. X ) ) -> ( T ` ( ( 1 ( .sOLD ` U ) A ) G B ) ) = ( ( 1 ( .sOLD ` W ) ( T ` A ) ) H ( T ` B ) ) )
10	1, 9	mp3anr1 1421	. 2 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( A e. X /\ B e. X ) ) -> ( T ` ( ( 1 ( .sOLD ` U ) A ) G B ) ) = ( ( 1 ( .sOLD ` W ) ( T ` A ) ) H ( T ` B ) ) )
11		simp1 1061	. . . . 5 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> U e. NrmCVec )
12		simpl 473	. . . . 5 |- ( ( A e. X /\ B e. X ) -> A e. X )
13	2, 6	nvsid 27482	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X ) -> ( 1 ( .sOLD ` U ) A ) = A )
14	11, 12, 13	syl2an 494	. . . 4 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( A e. X /\ B e. X ) ) -> ( 1 ( .sOLD ` U ) A ) = A )
15	14	oveq1d 6665	. . 3 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( A e. X /\ B e. X ) ) -> ( ( 1 ( .sOLD ` U ) A ) G B ) = ( A G B ) )
16	15	fveq2d 6195	. 2 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( A e. X /\ B e. X ) ) -> ( T ` ( ( 1 ( .sOLD ` U ) A ) G B ) ) = ( T ` ( A G B ) ) )
17		simpl2 1065	. . . 4 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( A e. X /\ B e. X ) ) -> W e. NrmCVec )
18	2, 3, 8	lnof 27610	. . . . 5 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> T : X --> ( BaseSet ` W ) )
19		ffvelrn 6357	. . . . 5 |- ( ( T : X --> ( BaseSet ` W ) /\ A e. X ) -> ( T ` A ) e. ( BaseSet ` W ) )
20	18, 12, 19	syl2an 494	. . . 4 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( A e. X /\ B e. X ) ) -> ( T ` A ) e. ( BaseSet ` W ) )
21	3, 7	nvsid 27482	. . . 4 |- ( ( W e. NrmCVec /\ ( T ` A ) e. ( BaseSet ` W ) ) -> ( 1 ( .sOLD ` W ) ( T ` A ) ) = ( T ` A ) )
22	17, 20, 21	syl2anc 693	. . 3 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( A e. X /\ B e. X ) ) -> ( 1 ( .sOLD ` W ) ( T ` A ) ) = ( T ` A ) )
23	22	oveq1d 6665	. 2 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( A e. X /\ B e. X ) ) -> ( ( 1 ( .sOLD ` W ) ( T ` A ) ) H ( T ` B ) ) = ( ( T ` A ) H ( T ` B ) ) )
24	10, 16, 23	3eqtr3d 2664	1 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( A e. X /\ B e. X ) ) -> ( T ` ( A G B ) ) = ( ( T ` A ) H ( T ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   LnOp clno 27595

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  ax-1cn 9994

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-pw 4160  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-1st 7168  df-2nd 7169  df-map 7859  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-lno 27599

This theorem is referenced by: (None)