Theorem lnopeq0lem1 28864

Description: Lemma for lnopeq0i 28866. Apply the generalized polarization identity polid2i 28014 to the quadratic form ( ( T ` x ) , x ). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnopeq0.1	|- T e. LinOp
lnopeq0lem1.2	|- A e. ~H
lnopeq0lem1.3	|- B e. ~H

Assertion

Ref	Expression
lnopeq0lem1	|- ( ( T ` A ) .ih B ) = ( ( ( ( ( T ` ( A +h B ) ) .ih ( A +h B ) ) - ( ( T ` ( A -h B ) ) .ih ( A -h B ) ) ) + ( _i x. ( ( ( T ` ( A +h ( _i .h B ) ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( T ` ( A -h ( _i .h B ) ) ) .ih ( A -h ( _i .h B ) ) ) ) ) ) / 4 )

Proof of Theorem lnopeq0lem1

Step	Hyp	Ref	Expression
1		lnopeq0lem1.2	. . . 4 |- A e. ~H
2		lnopeq0.1	. . . . . 6 |- T e. LinOp
3	2	lnopfi 28828	. . . . 5 |- T : ~H --> ~H
4	3	ffvelrni 6358	. . . 4 |- ( A e. ~H -> ( T ` A ) e. ~H )
5	1, 4	ax-mp 5	. . 3 |- ( T ` A ) e. ~H
6		lnopeq0lem1.3	. . 3 |- B e. ~H
7	3	ffvelrni 6358	. . . 4 |- ( B e. ~H -> ( T ` B ) e. ~H )
8	6, 7	ax-mp 5	. . 3 |- ( T ` B ) e. ~H
9	5, 6, 8, 1	polid2i 28014	. 2 |- ( ( T ` A ) .ih B ) = ( ( ( ( ( ( T ` A ) +h ( T ` B ) ) .ih ( A +h B ) ) - ( ( ( T ` A ) -h ( T ` B ) ) .ih ( A -h B ) ) ) + ( _i x. ( ( ( ( T ` A ) +h ( _i .h ( T ` B ) ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( ( T ` A ) -h ( _i .h ( T ` B ) ) ) .ih ( A -h ( _i .h B ) ) ) ) ) ) / 4 )
10	2	lnopaddi 28830	. . . . . . 7 |- ( ( A e. ~H /\ B e. ~H ) -> ( T ` ( A +h B ) ) = ( ( T ` A ) +h ( T ` B ) ) )
11	1, 6, 10	mp2an 708	. . . . . 6 |- ( T ` ( A +h B ) ) = ( ( T ` A ) +h ( T ` B ) )
12	11	oveq1i 6660	. . . . 5 |- ( ( T ` ( A +h B ) ) .ih ( A +h B ) ) = ( ( ( T ` A ) +h ( T ` B ) ) .ih ( A +h B ) )
13	2	lnopsubi 28833	. . . . . . 7 |- ( ( A e. ~H /\ B e. ~H ) -> ( T ` ( A -h B ) ) = ( ( T ` A ) -h ( T ` B ) ) )
14	1, 6, 13	mp2an 708	. . . . . 6 |- ( T ` ( A -h B ) ) = ( ( T ` A ) -h ( T ` B ) )
15	14	oveq1i 6660	. . . . 5 |- ( ( T ` ( A -h B ) ) .ih ( A -h B ) ) = ( ( ( T ` A ) -h ( T ` B ) ) .ih ( A -h B ) )
16	12, 15	oveq12i 6662	. . . 4 |- ( ( ( T ` ( A +h B ) ) .ih ( A +h B ) ) - ( ( T ` ( A -h B ) ) .ih ( A -h B ) ) ) = ( ( ( ( T ` A ) +h ( T ` B ) ) .ih ( A +h B ) ) - ( ( ( T ` A ) -h ( T ` B ) ) .ih ( A -h B ) ) )
17		ax-icn 9995	. . . . . . . 8 |- _i e. CC
18	2	lnopaddmuli 28832	. . . . . . . 8 |- ( ( _i e. CC /\ A e. ~H /\ B e. ~H ) -> ( T ` ( A +h ( _i .h B ) ) ) = ( ( T ` A ) +h ( _i .h ( T ` B ) ) ) )
19	17, 1, 6, 18	mp3an 1424	. . . . . . 7 |- ( T ` ( A +h ( _i .h B ) ) ) = ( ( T ` A ) +h ( _i .h ( T ` B ) ) )
20	19	oveq1i 6660	. . . . . 6 |- ( ( T ` ( A +h ( _i .h B ) ) ) .ih ( A +h ( _i .h B ) ) ) = ( ( ( T ` A ) +h ( _i .h ( T ` B ) ) ) .ih ( A +h ( _i .h B ) ) )
21	2	lnopsubmuli 28834	. . . . . . . 8 |- ( ( _i e. CC /\ A e. ~H /\ B e. ~H ) -> ( T ` ( A -h ( _i .h B ) ) ) = ( ( T ` A ) -h ( _i .h ( T ` B ) ) ) )
22	17, 1, 6, 21	mp3an 1424	. . . . . . 7 |- ( T ` ( A -h ( _i .h B ) ) ) = ( ( T ` A ) -h ( _i .h ( T ` B ) ) )
23	22	oveq1i 6660	. . . . . 6 |- ( ( T ` ( A -h ( _i .h B ) ) ) .ih ( A -h ( _i .h B ) ) ) = ( ( ( T ` A ) -h ( _i .h ( T ` B ) ) ) .ih ( A -h ( _i .h B ) ) )
24	20, 23	oveq12i 6662	. . . . 5 |- ( ( ( T ` ( A +h ( _i .h B ) ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( T ` ( A -h ( _i .h B ) ) ) .ih ( A -h ( _i .h B ) ) ) ) = ( ( ( ( T ` A ) +h ( _i .h ( T ` B ) ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( ( T ` A ) -h ( _i .h ( T ` B ) ) ) .ih ( A -h ( _i .h B ) ) ) )
25	24	oveq2i 6661	. . . 4 |- ( _i x. ( ( ( T ` ( A +h ( _i .h B ) ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( T ` ( A -h ( _i .h B ) ) ) .ih ( A -h ( _i .h B ) ) ) ) ) = ( _i x. ( ( ( ( T ` A ) +h ( _i .h ( T ` B ) ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( ( T ` A ) -h ( _i .h ( T ` B ) ) ) .ih ( A -h ( _i .h B ) ) ) ) )
26	16, 25	oveq12i 6662	. . 3 |- ( ( ( ( T ` ( A +h B ) ) .ih ( A +h B ) ) - ( ( T ` ( A -h B ) ) .ih ( A -h B ) ) ) + ( _i x. ( ( ( T ` ( A +h ( _i .h B ) ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( T ` ( A -h ( _i .h B ) ) ) .ih ( A -h ( _i .h B ) ) ) ) ) ) = ( ( ( ( ( T ` A ) +h ( T ` B ) ) .ih ( A +h B ) ) - ( ( ( T ` A ) -h ( T ` B ) ) .ih ( A -h B ) ) ) + ( _i x. ( ( ( ( T ` A ) +h ( _i .h ( T ` B ) ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( ( T ` A ) -h ( _i .h ( T ` B ) ) ) .ih ( A -h ( _i .h B ) ) ) ) ) )
27	26	oveq1i 6660	. 2 |- ( ( ( ( ( T ` ( A +h B ) ) .ih ( A +h B ) ) - ( ( T ` ( A -h B ) ) .ih ( A -h B ) ) ) + ( _i x. ( ( ( T ` ( A +h ( _i .h B ) ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( T ` ( A -h ( _i .h B ) ) ) .ih ( A -h ( _i .h B ) ) ) ) ) ) / 4 ) = ( ( ( ( ( ( T ` A ) +h ( T ` B ) ) .ih ( A +h B ) ) - ( ( ( T ` A ) -h ( T ` B ) ) .ih ( A -h B ) ) ) + ( _i x. ( ( ( ( T ` A ) +h ( _i .h ( T ` B ) ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( ( T ` A ) -h ( _i .h ( T ` B ) ) ) .ih ( A -h ( _i .h B ) ) ) ) ) ) / 4 )
28	9, 27	eqtr4i 2647	1 |- ( ( T ` A ) .ih B ) = ( ( ( ( ( T ` ( A +h B ) ) .ih ( A +h B ) ) - ( ( T ` ( A -h B ) ) .ih ( A -h B ) ) ) + ( _i x. ( ( ( T ` ( A +h ( _i .h B ) ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( T ` ( A -h ( _i .h B ) ) ) .ih ( A -h ( _i .h B ) ) ) ) ) ) / 4 )

Syntax hints:   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   4c4 11072   ~Hchil 27776   +h cva 27777   .h csm 27778   .ih csp 27779   -h cmv 27782   LinOpclo 27804

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-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-hilex 27856  ax-hfvadd 27857  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-po 5035  df-so 5036  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-3 11080  df-4 11081  df-cj 13839  df-re 13840  df-im 13841  df-hvsub 27828  df-lnop 28700

This theorem is referenced by:  lnopeq0lem2  28865