Theorem bralnfn 28807

Description: The Dirac bra function is a linear functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
bralnfn	|- ( A e. ~H -> ( bra ` A ) e. LinFn )

Proof of Theorem bralnfn

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

Step	Hyp	Ref	Expression
1		brafn 28806	. 2 |- ( A e. ~H -> ( bra ` A ) : ~H --> CC )
2		simpll 790	. . . . . 6 |- ( ( ( A e. ~H /\ x e. CC ) /\ ( y e. ~H /\ z e. ~H ) ) -> A e. ~H )
3		hvmulcl 27870	. . . . . . 7 |- ( ( x e. CC /\ y e. ~H ) -> ( x .h y ) e. ~H )
4	3	ad2ant2lr 784	. . . . . 6 |- ( ( ( A e. ~H /\ x e. CC ) /\ ( y e. ~H /\ z e. ~H ) ) -> ( x .h y ) e. ~H )
5		simprr 796	. . . . . 6 |- ( ( ( A e. ~H /\ x e. CC ) /\ ( y e. ~H /\ z e. ~H ) ) -> z e. ~H )
6		braadd 28804	. . . . . 6 |- ( ( A e. ~H /\ ( x .h y ) e. ~H /\ z e. ~H ) -> ( ( bra ` A ) ` ( ( x .h y ) +h z ) ) = ( ( ( bra ` A ) ` ( x .h y ) ) + ( ( bra ` A ) ` z ) ) )
7	2, 4, 5, 6	syl3anc 1326	. . . . 5 |- ( ( ( A e. ~H /\ x e. CC ) /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( bra ` A ) ` ( ( x .h y ) +h z ) ) = ( ( ( bra ` A ) ` ( x .h y ) ) + ( ( bra ` A ) ` z ) ) )
8		bramul 28805	. . . . . . . 8 |- ( ( A e. ~H /\ x e. CC /\ y e. ~H ) -> ( ( bra ` A ) ` ( x .h y ) ) = ( x x. ( ( bra ` A ) ` y ) ) )
9	8	3expa 1265	. . . . . . 7 |- ( ( ( A e. ~H /\ x e. CC ) /\ y e. ~H ) -> ( ( bra ` A ) ` ( x .h y ) ) = ( x x. ( ( bra ` A ) ` y ) ) )
10	9	adantrr 753	. . . . . 6 |- ( ( ( A e. ~H /\ x e. CC ) /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( bra ` A ) ` ( x .h y ) ) = ( x x. ( ( bra ` A ) ` y ) ) )
11	10	oveq1d 6665	. . . . 5 |- ( ( ( A e. ~H /\ x e. CC ) /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( ( bra ` A ) ` ( x .h y ) ) + ( ( bra ` A ) ` z ) ) = ( ( x x. ( ( bra ` A ) ` y ) ) + ( ( bra ` A ) ` z ) ) )
12	7, 11	eqtrd 2656	. . . 4 |- ( ( ( A e. ~H /\ x e. CC ) /\ ( y e. ~H /\ z e. ~H ) ) -> ( ( bra ` A ) ` ( ( x .h y ) +h z ) ) = ( ( x x. ( ( bra ` A ) ` y ) ) + ( ( bra ` A ) ` z ) ) )
13	12	ralrimivva 2971	. . 3 |- ( ( A e. ~H /\ x e. CC ) -> A. y e. ~H A. z e. ~H ( ( bra ` A ) ` ( ( x .h y ) +h z ) ) = ( ( x x. ( ( bra ` A ) ` y ) ) + ( ( bra ` A ) ` z ) ) )
14	13	ralrimiva 2966	. 2 |- ( A e. ~H -> A. x e. CC A. y e. ~H A. z e. ~H ( ( bra ` A ) ` ( ( x .h y ) +h z ) ) = ( ( x x. ( ( bra ` A ) ` y ) ) + ( ( bra ` A ) ` z ) ) )
15		ellnfn 28742	. 2 |- ( ( bra ` A ) e. LinFn <-> ( ( bra ` A ) : ~H --> CC /\ A. x e. CC A. y e. ~H A. z e. ~H ( ( bra ` A ) ` ( ( x .h y ) +h z ) ) = ( ( x x. ( ( bra ` A ) ` y ) ) + ( ( bra ` A ) ` z ) ) ) )
16	1, 14, 15	sylanbrc 698	1 |- ( A e. ~H -> ( bra ` A ) e. LinFn )

Syntax hints:   -> wi 4   /\ wa 384   = 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   bracbr 27813

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-cnex 9992  ax-hilex 27856  ax-hfvadd 27857  ax-hfvmul 27862  ax-hfi 27936  ax-his2 27940  ax-his3 27941

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-map 7859  df-lnfn 28707  df-bra 28709

This theorem is referenced by:  rnbra  28966  kbass4  28978