Theorem normpari 28011

Description: Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
normpar.1	|- A e. ~H
normpar.2	|- B e. ~H

Assertion

Ref	Expression
normpari	|- ( ( ( normh ` ( A -h B ) ) ^ 2 ) + ( ( normh ` ( A +h B ) ) ^ 2 ) ) = ( ( 2 x. ( ( normh ` A ) ^ 2 ) ) + ( 2 x. ( ( normh ` B ) ^ 2 ) ) )

Proof of Theorem normpari

Step	Hyp	Ref	Expression
1		normpar.1	. . . . 5 |- A e. ~H
2		normpar.2	. . . . 5 |- B e. ~H
3	1, 2	hvsubcli 27878	. . . 4 |- ( A -h B ) e. ~H
4	3	normsqi 27989	. . 3 |- ( ( normh ` ( A -h B ) ) ^ 2 ) = ( ( A -h B ) .ih ( A -h B ) )
5	1, 2	hvaddcli 27875	. . . 4 |- ( A +h B ) e. ~H
6	5	normsqi 27989	. . 3 |- ( ( normh ` ( A +h B ) ) ^ 2 ) = ( ( A +h B ) .ih ( A +h B ) )
7	4, 6	oveq12i 6662	. 2 |- ( ( ( normh ` ( A -h B ) ) ^ 2 ) + ( ( normh ` ( A +h B ) ) ^ 2 ) ) = ( ( ( A -h B ) .ih ( A -h B ) ) + ( ( A +h B ) .ih ( A +h B ) ) )
8	1	normsqi 27989	. . . . . 6 |- ( ( normh ` A ) ^ 2 ) = ( A .ih A )
9	8	oveq2i 6661	. . . . 5 |- ( 2 x. ( ( normh ` A ) ^ 2 ) ) = ( 2 x. ( A .ih A ) )
10	1, 1	hicli 27938	. . . . . 6 |- ( A .ih A ) e. CC
11	10	2timesi 11147	. . . . 5 |- ( 2 x. ( A .ih A ) ) = ( ( A .ih A ) + ( A .ih A ) )
12	9, 11	eqtri 2644	. . . 4 |- ( 2 x. ( ( normh ` A ) ^ 2 ) ) = ( ( A .ih A ) + ( A .ih A ) )
13	2	normsqi 27989	. . . . . 6 |- ( ( normh ` B ) ^ 2 ) = ( B .ih B )
14	13	oveq2i 6661	. . . . 5 |- ( 2 x. ( ( normh ` B ) ^ 2 ) ) = ( 2 x. ( B .ih B ) )
15	2, 2	hicli 27938	. . . . . 6 |- ( B .ih B ) e. CC
16	15	2timesi 11147	. . . . 5 |- ( 2 x. ( B .ih B ) ) = ( ( B .ih B ) + ( B .ih B ) )
17	14, 16	eqtri 2644	. . . 4 |- ( 2 x. ( ( normh ` B ) ^ 2 ) ) = ( ( B .ih B ) + ( B .ih B ) )
18	12, 17	oveq12i 6662	. . 3 |- ( ( 2 x. ( ( normh ` A ) ^ 2 ) ) + ( 2 x. ( ( normh ` B ) ^ 2 ) ) ) = ( ( ( A .ih A ) + ( A .ih A ) ) + ( ( B .ih B ) + ( B .ih B ) ) )
19	1, 2, 1, 2	normlem9 27975	. . . . . 6 |- ( ( A -h B ) .ih ( A -h B ) ) = ( ( ( A .ih A ) + ( B .ih B ) ) - ( ( A .ih B ) + ( B .ih A ) ) )
20	10, 15	addcli 10044	. . . . . . 7 |- ( ( A .ih A ) + ( B .ih B ) ) e. CC
21	1, 2	hicli 27938	. . . . . . . 8 |- ( A .ih B ) e. CC
22	2, 1	hicli 27938	. . . . . . . 8 |- ( B .ih A ) e. CC
23	21, 22	addcli 10044	. . . . . . 7 |- ( ( A .ih B ) + ( B .ih A ) ) e. CC
24	20, 23	negsubi 10359	. . . . . 6 |- ( ( ( A .ih A ) + ( B .ih B ) ) + -u ( ( A .ih B ) + ( B .ih A ) ) ) = ( ( ( A .ih A ) + ( B .ih B ) ) - ( ( A .ih B ) + ( B .ih A ) ) )
25	19, 24	eqtr4i 2647	. . . . 5 |- ( ( A -h B ) .ih ( A -h B ) ) = ( ( ( A .ih A ) + ( B .ih B ) ) + -u ( ( A .ih B ) + ( B .ih A ) ) )
26	1, 2, 1, 2	normlem8 27974	. . . . 5 |- ( ( A +h B ) .ih ( A +h B ) ) = ( ( ( A .ih A ) + ( B .ih B ) ) + ( ( A .ih B ) + ( B .ih A ) ) )
27	25, 26	oveq12i 6662	. . . 4 |- ( ( ( A -h B ) .ih ( A -h B ) ) + ( ( A +h B ) .ih ( A +h B ) ) ) = ( ( ( ( A .ih A ) + ( B .ih B ) ) + -u ( ( A .ih B ) + ( B .ih A ) ) ) + ( ( ( A .ih A ) + ( B .ih B ) ) + ( ( A .ih B ) + ( B .ih A ) ) ) )
28	23	negcli 10349	. . . . 5 |- -u ( ( A .ih B ) + ( B .ih A ) ) e. CC
29	20, 28, 20, 23	add42i 10261	. . . 4 |- ( ( ( ( A .ih A ) + ( B .ih B ) ) + -u ( ( A .ih B ) + ( B .ih A ) ) ) + ( ( ( A .ih A ) + ( B .ih B ) ) + ( ( A .ih B ) + ( B .ih A ) ) ) ) = ( ( ( ( A .ih A ) + ( B .ih B ) ) + ( ( A .ih A ) + ( B .ih B ) ) ) + ( ( ( A .ih B ) + ( B .ih A ) ) + -u ( ( A .ih B ) + ( B .ih A ) ) ) )
30	23	negidi 10350	. . . . . 6 |- ( ( ( A .ih B ) + ( B .ih A ) ) + -u ( ( A .ih B ) + ( B .ih A ) ) ) = 0
31	30	oveq2i 6661	. . . . 5 |- ( ( ( ( A .ih A ) + ( B .ih B ) ) + ( ( A .ih A ) + ( B .ih B ) ) ) + ( ( ( A .ih B ) + ( B .ih A ) ) + -u ( ( A .ih B ) + ( B .ih A ) ) ) ) = ( ( ( ( A .ih A ) + ( B .ih B ) ) + ( ( A .ih A ) + ( B .ih B ) ) ) + 0 )
32	20, 20	addcli 10044	. . . . . 6 |- ( ( ( A .ih A ) + ( B .ih B ) ) + ( ( A .ih A ) + ( B .ih B ) ) ) e. CC
33	32	addid1i 10223	. . . . 5 |- ( ( ( ( A .ih A ) + ( B .ih B ) ) + ( ( A .ih A ) + ( B .ih B ) ) ) + 0 ) = ( ( ( A .ih A ) + ( B .ih B ) ) + ( ( A .ih A ) + ( B .ih B ) ) )
34	10, 15, 10, 15	add4i 10260	. . . . 5 |- ( ( ( A .ih A ) + ( B .ih B ) ) + ( ( A .ih A ) + ( B .ih B ) ) ) = ( ( ( A .ih A ) + ( A .ih A ) ) + ( ( B .ih B ) + ( B .ih B ) ) )
35	31, 33, 34	3eqtri 2648	. . . 4 |- ( ( ( ( A .ih A ) + ( B .ih B ) ) + ( ( A .ih A ) + ( B .ih B ) ) ) + ( ( ( A .ih B ) + ( B .ih A ) ) + -u ( ( A .ih B ) + ( B .ih A ) ) ) ) = ( ( ( A .ih A ) + ( A .ih A ) ) + ( ( B .ih B ) + ( B .ih B ) ) )
36	27, 29, 35	3eqtri 2648	. . 3 |- ( ( ( A -h B ) .ih ( A -h B ) ) + ( ( A +h B ) .ih ( A +h B ) ) ) = ( ( ( A .ih A ) + ( A .ih A ) ) + ( ( B .ih B ) + ( B .ih B ) ) )
37	18, 36	eqtr4i 2647	. 2 |- ( ( 2 x. ( ( normh ` A ) ^ 2 ) ) + ( 2 x. ( ( normh ` B ) ^ 2 ) ) ) = ( ( ( A -h B ) .ih ( A -h B ) ) + ( ( A +h B ) .ih ( A +h B ) ) )
38	7, 37	eqtr4i 2647	1 |- ( ( ( normh ` ( A -h B ) ) ^ 2 ) + ( ( normh ` ( A +h B ) ) ^ 2 ) ) = ( ( 2 x. ( ( normh ` A ) ^ 2 ) ) + ( 2 x. ( ( normh ` B ) ^ 2 ) ) )

Syntax hints:   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   0cc0 9936   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   2c2 11070   ^cexp 12860   ~Hchil 27776   +h cva 27777   .ih csp 27779   normhcno 27780   -h cmv 27782

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-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-pre-sup 10014  ax-hfvadd 27857  ax-hv0cl 27860  ax-hfvmul 27862  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942

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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  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-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-hnorm 27825  df-hvsub 27828

This theorem is referenced by:  normpar  28012  normpar2i  28013