Theorem lnfnl 28790

Description: Basic linearity property of a linear functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
lnfnl	|- ( ( ( T e. LinFn /\ A e. CC ) /\ ( B e. ~H /\ C e. ~H ) ) -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) )

Proof of Theorem lnfnl

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ellnfn 28742	. . . . . 6 |- ( T e. LinFn <-> ( T : ~H --> CC /\ A. x e. CC A. y e. ~H A. z e. ~H ( T ` ( ( x .h y ) +h z ) ) = ( ( x x. ( T ` y ) ) + ( T ` z ) ) ) )
2	1	simprbi 480	. . . . 5 |- ( T e. LinFn -> A. x e. CC A. y e. ~H A. z e. ~H ( T ` ( ( x .h y ) +h z ) ) = ( ( x x. ( T ` y ) ) + ( T ` z ) ) )
3		oveq1 6657	. . . . . . . . 9 |- ( x = A -> ( x .h y ) = ( A .h y ) )
4	3	oveq1d 6665	. . . . . . . 8 |- ( x = A -> ( ( x .h y ) +h z ) = ( ( A .h y ) +h z ) )
5	4	fveq2d 6195	. . . . . . 7 |- ( x = A -> ( T ` ( ( x .h y ) +h z ) ) = ( T ` ( ( A .h y ) +h z ) ) )
6		oveq1 6657	. . . . . . . 8 |- ( x = A -> ( x x. ( T ` y ) ) = ( A x. ( T ` y ) ) )
7	6	oveq1d 6665	. . . . . . 7 |- ( x = A -> ( ( x x. ( T ` y ) ) + ( T ` z ) ) = ( ( A x. ( T ` y ) ) + ( T ` z ) ) )
8	5, 7	eqeq12d 2637	. . . . . 6 |- ( x = A -> ( ( T ` ( ( x .h y ) +h z ) ) = ( ( x x. ( T ` y ) ) + ( T ` z ) ) <-> ( T ` ( ( A .h y ) +h z ) ) = ( ( A x. ( T ` y ) ) + ( T ` z ) ) ) )
9		oveq2 6658	. . . . . . . . 9 |- ( y = B -> ( A .h y ) = ( A .h B ) )
10	9	oveq1d 6665	. . . . . . . 8 |- ( y = B -> ( ( A .h y ) +h z ) = ( ( A .h B ) +h z ) )
11	10	fveq2d 6195	. . . . . . 7 |- ( y = B -> ( T ` ( ( A .h y ) +h z ) ) = ( T ` ( ( A .h B ) +h z ) ) )
12		fveq2 6191	. . . . . . . . 9 |- ( y = B -> ( T ` y ) = ( T ` B ) )
13	12	oveq2d 6666	. . . . . . . 8 |- ( y = B -> ( A x. ( T ` y ) ) = ( A x. ( T ` B ) ) )
14	13	oveq1d 6665	. . . . . . 7 |- ( y = B -> ( ( A x. ( T ` y ) ) + ( T ` z ) ) = ( ( A x. ( T ` B ) ) + ( T ` z ) ) )
15	11, 14	eqeq12d 2637	. . . . . 6 |- ( y = B -> ( ( T ` ( ( A .h y ) +h z ) ) = ( ( A x. ( T ` y ) ) + ( T ` z ) ) <-> ( T ` ( ( A .h B ) +h z ) ) = ( ( A x. ( T ` B ) ) + ( T ` z ) ) ) )
16		oveq2 6658	. . . . . . . 8 |- ( z = C -> ( ( A .h B ) +h z ) = ( ( A .h B ) +h C ) )
17	16	fveq2d 6195	. . . . . . 7 |- ( z = C -> ( T ` ( ( A .h B ) +h z ) ) = ( T ` ( ( A .h B ) +h C ) ) )
18		fveq2 6191	. . . . . . . 8 |- ( z = C -> ( T ` z ) = ( T ` C ) )
19	18	oveq2d 6666	. . . . . . 7 |- ( z = C -> ( ( A x. ( T ` B ) ) + ( T ` z ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) )
20	17, 19	eqeq12d 2637	. . . . . 6 |- ( z = C -> ( ( T ` ( ( A .h B ) +h z ) ) = ( ( A x. ( T ` B ) ) + ( T ` z ) ) <-> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) ) )
21	8, 15, 20	rspc3v 3325	. . . . 5 |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( A. x e. CC A. y e. ~H A. z e. ~H ( T ` ( ( x .h y ) +h z ) ) = ( ( x x. ( T ` y ) ) + ( T ` z ) ) -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) ) )
22	2, 21	syl5 34	. . . 4 |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( T e. LinFn -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) ) )
23	22	3expb 1266	. . 3 |- ( ( A e. CC /\ ( B e. ~H /\ C e. ~H ) ) -> ( T e. LinFn -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) ) )
24	23	impcom 446	. 2 |- ( ( T e. LinFn /\ ( A e. CC /\ ( B e. ~H /\ C e. ~H ) ) ) -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) )
25	24	anassrs 680	1 |- ( ( ( T e. LinFn /\ A e. CC ) /\ ( B e. ~H /\ C e. ~H ) ) -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( 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   + caddc 9939   x. cmul 9941   ~Hchil 27776   +h cva 27777   .h csm 27778   LinFnclf 27811

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  ax-cnex 9992  ax-hilex 27856

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-lnfn 28707

This theorem is referenced by:  lnfnli  28899