Theorem ellnfn 28742

Description: Property defining a linear functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ellnfn	|- ( 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 ) ) ) )

Distinct variable group:   x, y, z, T

Proof of Theorem ellnfn

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6190	. . . . . 6 |- ( t = T -> ( t ` ( ( x .h y ) +h z ) ) = ( T ` ( ( x .h y ) +h z ) ) )
2		fveq1 6190	. . . . . . . 8 |- ( t = T -> ( t ` y ) = ( T ` y ) )
3	2	oveq2d 6666	. . . . . . 7 |- ( t = T -> ( x x. ( t ` y ) ) = ( x x. ( T ` y ) ) )
4		fveq1 6190	. . . . . . 7 |- ( t = T -> ( t ` z ) = ( T ` z ) )
5	3, 4	oveq12d 6668	. . . . . 6 |- ( t = T -> ( ( x x. ( t ` y ) ) + ( t ` z ) ) = ( ( x x. ( T ` y ) ) + ( T ` z ) ) )
6	1, 5	eqeq12d 2637	. . . . 5 |- ( t = T -> ( ( t ` ( ( x .h y ) +h z ) ) = ( ( x x. ( t ` y ) ) + ( t ` z ) ) <-> ( T ` ( ( x .h y ) +h z ) ) = ( ( x x. ( T ` y ) ) + ( T ` z ) ) ) )
7	6	ralbidv 2986	. . . 4 |- ( t = T -> ( A. z e. ~H ( t ` ( ( x .h y ) +h z ) ) = ( ( x x. ( t ` y ) ) + ( t ` z ) ) <-> A. z e. ~H ( T ` ( ( x .h y ) +h z ) ) = ( ( x x. ( T ` y ) ) + ( T ` z ) ) ) )
8	7	2ralbidv 2989	. . 3 |- ( t = T -> ( A. x e. CC A. y e. ~H A. z e. ~H ( t ` ( ( x .h y ) +h z ) ) = ( ( x x. ( t ` y ) ) + ( t ` z ) ) <-> A. x e. CC A. y e. ~H A. z e. ~H ( T ` ( ( x .h y ) +h z ) ) = ( ( x x. ( T ` y ) ) + ( T ` z ) ) ) )
9		df-lnfn 28707	. . 3 |- LinFn = { t e. ( CC ^m ~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 ) ) }
10	8, 9	elrab2 3366	. 2 |- ( T e. LinFn <-> ( T e. ( CC ^m ~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 ) ) ) )
11		cnex 10017	. . . 4 |- CC e. _V
12		ax-hilex 27856	. . . 4 |- ~H e. _V
13	11, 12	elmap 7886	. . 3 |- ( T e. ( CC ^m ~H ) <-> T : ~H --> CC )
14	13	anbi1i 731	. 2 |- ( ( T e. ( CC ^m ~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 : ~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 ) ) ) )
15	10, 14	bitri 264	1 |- ( 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 ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   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:  lnfnf  28743  lnfnl  28790  bralnfn  28807  0lnfn  28844  cnlnadjlem2  28927