Theorem lnolin 27609

Description: Basic linearity property of a linear operator. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnoval.1	|- X = ( BaseSet ` U )
lnoval.2	|- Y = ( BaseSet ` W )
lnoval.3	|- G = ( +v ` U )
lnoval.4	|- H = ( +v ` W )
lnoval.5	|- R = ( .sOLD ` U )
lnoval.6	|- S = ( .sOLD ` W )
lnoval.7	|- L = ( U LnOp W )

Assertion

Ref	Expression
lnolin	|- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( T ` ( ( A R B ) G C ) ) = ( ( A S ( T ` B ) ) H ( T ` C ) ) )

Proof of Theorem lnolin

Dummy variables u t w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnoval.1	. . . . 5 |- X = ( BaseSet ` U )
2		lnoval.2	. . . . 5 |- Y = ( BaseSet ` W )
3		lnoval.3	. . . . 5 |- G = ( +v ` U )
4		lnoval.4	. . . . 5 |- H = ( +v ` W )
5		lnoval.5	. . . . 5 |- R = ( .sOLD ` U )
6		lnoval.6	. . . . 5 |- S = ( .sOLD ` W )
7		lnoval.7	. . . . 5 |- L = ( U LnOp W )
8	1, 2, 3, 4, 5, 6, 7	islno 27608	. . . 4 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( T e. L <-> ( T : X --> Y /\ A. u e. CC A. w e. X A. t e. X ( T ` ( ( u R w ) G t ) ) = ( ( u S ( T ` w ) ) H ( T ` t ) ) ) ) )
9	8	biimp3a 1432	. . 3 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> ( T : X --> Y /\ A. u e. CC A. w e. X A. t e. X ( T ` ( ( u R w ) G t ) ) = ( ( u S ( T ` w ) ) H ( T ` t ) ) ) )
10	9	simprd 479	. 2 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) -> A. u e. CC A. w e. X A. t e. X ( T ` ( ( u R w ) G t ) ) = ( ( u S ( T ` w ) ) H ( T ` t ) ) )
11		oveq1 6657	. . . . . 6 |- ( u = A -> ( u R w ) = ( A R w ) )
12	11	oveq1d 6665	. . . . 5 |- ( u = A -> ( ( u R w ) G t ) = ( ( A R w ) G t ) )
13	12	fveq2d 6195	. . . 4 |- ( u = A -> ( T ` ( ( u R w ) G t ) ) = ( T ` ( ( A R w ) G t ) ) )
14		oveq1 6657	. . . . 5 |- ( u = A -> ( u S ( T ` w ) ) = ( A S ( T ` w ) ) )
15	14	oveq1d 6665	. . . 4 |- ( u = A -> ( ( u S ( T ` w ) ) H ( T ` t ) ) = ( ( A S ( T ` w ) ) H ( T ` t ) ) )
16	13, 15	eqeq12d 2637	. . 3 |- ( u = A -> ( ( T ` ( ( u R w ) G t ) ) = ( ( u S ( T ` w ) ) H ( T ` t ) ) <-> ( T ` ( ( A R w ) G t ) ) = ( ( A S ( T ` w ) ) H ( T ` t ) ) ) )
17		oveq2 6658	. . . . . 6 |- ( w = B -> ( A R w ) = ( A R B ) )
18	17	oveq1d 6665	. . . . 5 |- ( w = B -> ( ( A R w ) G t ) = ( ( A R B ) G t ) )
19	18	fveq2d 6195	. . . 4 |- ( w = B -> ( T ` ( ( A R w ) G t ) ) = ( T ` ( ( A R B ) G t ) ) )
20		fveq2 6191	. . . . . 6 |- ( w = B -> ( T ` w ) = ( T ` B ) )
21	20	oveq2d 6666	. . . . 5 |- ( w = B -> ( A S ( T ` w ) ) = ( A S ( T ` B ) ) )
22	21	oveq1d 6665	. . . 4 |- ( w = B -> ( ( A S ( T ` w ) ) H ( T ` t ) ) = ( ( A S ( T ` B ) ) H ( T ` t ) ) )
23	19, 22	eqeq12d 2637	. . 3 |- ( w = B -> ( ( T ` ( ( A R w ) G t ) ) = ( ( A S ( T ` w ) ) H ( T ` t ) ) <-> ( T ` ( ( A R B ) G t ) ) = ( ( A S ( T ` B ) ) H ( T ` t ) ) ) )
24		oveq2 6658	. . . . 5 |- ( t = C -> ( ( A R B ) G t ) = ( ( A R B ) G C ) )
25	24	fveq2d 6195	. . . 4 |- ( t = C -> ( T ` ( ( A R B ) G t ) ) = ( T ` ( ( A R B ) G C ) ) )
26		fveq2 6191	. . . . 5 |- ( t = C -> ( T ` t ) = ( T ` C ) )
27	26	oveq2d 6666	. . . 4 |- ( t = C -> ( ( A S ( T ` B ) ) H ( T ` t ) ) = ( ( A S ( T ` B ) ) H ( T ` C ) ) )
28	25, 27	eqeq12d 2637	. . 3 |- ( t = C -> ( ( T ` ( ( A R B ) G t ) ) = ( ( A S ( T ` B ) ) H ( T ` t ) ) <-> ( T ` ( ( A R B ) G C ) ) = ( ( A S ( T ` B ) ) H ( T ` C ) ) ) )
29	16, 23, 28	rspc3v 3325	. 2 |- ( ( A e. CC /\ B e. X /\ C e. X ) -> ( A. u e. CC A. w e. X A. t e. X ( T ` ( ( u R w ) G t ) ) = ( ( u S ( T ` w ) ) H ( T ` t ) ) -> ( T ` ( ( A R B ) G C ) ) = ( ( A S ( T ` B ) ) H ( T ` C ) ) ) )
30	10, 29	mpan9 486	1 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec /\ T e. L ) /\ ( A e. CC /\ B e. X /\ C e. X ) ) -> ( T ` ( ( A R B ) G C ) ) = ( ( A S ( T ` B ) ) H ( T ` C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   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-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-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-pw 4160  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-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-lno 27599

This theorem is referenced by:  lno0  27611  lnocoi  27612  lnoadd  27613  lnosub  27614  lnomul  27615