Theorem lnfnli 28899

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

Hypothesis

Ref	Expression
lnfnl.1	|- T e. LinFn

Assertion

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

Proof of Theorem lnfnli

Step	Hyp	Ref	Expression
1		lnfnl.1	. . 3 |- T e. LinFn
2		lnfnl 28790	. . 3 |- ( ( ( T e. LinFn /\ A e. CC ) /\ ( B e. ~H /\ C e. ~H ) ) -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) )
3	1, 2	mpanl1 716	. 2 |- ( ( A e. CC /\ ( B e. ~H /\ C e. ~H ) ) -> ( T ` ( ( A .h B ) +h C ) ) = ( ( A x. ( T ` B ) ) + ( T ` C ) ) )
4	3	3impb 1260	1 |- ( ( 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   ` 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:  lnfn0i  28901  lnfnaddi  28902  lnfnmuli  28903