Theorem normlem0 27966

Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 7-Oct-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
normlem1.1	|- S e. CC
normlem1.2	|- F e. ~H
normlem1.3	|- G e. ~H

Assertion

Ref	Expression
normlem0	|- ( ( F -h ( S .h G ) ) .ih ( F -h ( S .h G ) ) ) = ( ( ( F .ih F ) + ( -u ( * ` S ) x. ( F .ih G ) ) ) + ( ( -u S x. ( G .ih F ) ) + ( ( S x. ( * ` S ) ) x. ( G .ih G ) ) ) )

Proof of Theorem normlem0

Step	Hyp	Ref	Expression
1		normlem1.2	. . . . 5 |- F e. ~H
2		normlem1.1	. . . . . 6 |- S e. CC
3		normlem1.3	. . . . . 6 |- G e. ~H
4	2, 3	hvmulcli 27871	. . . . 5 |- ( S .h G ) e. ~H
5	1, 4	hvsubvali 27877	. . . 4 |- ( F -h ( S .h G ) ) = ( F +h ( -u 1 .h ( S .h G ) ) )
6	2	mulm1i 10475	. . . . . . 7 |- ( -u 1 x. S ) = -u S
7	6	oveq1i 6660	. . . . . 6 |- ( ( -u 1 x. S ) .h G ) = ( -u S .h G )
8		neg1cn 11124	. . . . . . 7 |- -u 1 e. CC
9	8, 2, 3	hvmulassi 27903	. . . . . 6 |- ( ( -u 1 x. S ) .h G ) = ( -u 1 .h ( S .h G ) )
10	7, 9	eqtr3i 2646	. . . . 5 |- ( -u S .h G ) = ( -u 1 .h ( S .h G ) )
11	10	oveq2i 6661	. . . 4 |- ( F +h ( -u S .h G ) ) = ( F +h ( -u 1 .h ( S .h G ) ) )
12	5, 11	eqtr4i 2647	. . 3 |- ( F -h ( S .h G ) ) = ( F +h ( -u S .h G ) )
13	12, 12	oveq12i 6662	. 2 |- ( ( F -h ( S .h G ) ) .ih ( F -h ( S .h G ) ) ) = ( ( F +h ( -u S .h G ) ) .ih ( F +h ( -u S .h G ) ) )
14	2	negcli 10349	. . . 4 |- -u S e. CC
15	14, 3	hvmulcli 27871	. . 3 |- ( -u S .h G ) e. ~H
16	1, 15	hvaddcli 27875	. . 3 |- ( F +h ( -u S .h G ) ) e. ~H
17		ax-his2 27940	. . 3 |- ( ( F e. ~H /\ ( -u S .h G ) e. ~H /\ ( F +h ( -u S .h G ) ) e. ~H ) -> ( ( F +h ( -u S .h G ) ) .ih ( F +h ( -u S .h G ) ) ) = ( ( F .ih ( F +h ( -u S .h G ) ) ) + ( ( -u S .h G ) .ih ( F +h ( -u S .h G ) ) ) ) )
18	1, 15, 16, 17	mp3an 1424	. 2 |- ( ( F +h ( -u S .h G ) ) .ih ( F +h ( -u S .h G ) ) ) = ( ( F .ih ( F +h ( -u S .h G ) ) ) + ( ( -u S .h G ) .ih ( F +h ( -u S .h G ) ) ) )
19		his7 27947	. . . . 5 |- ( ( F e. ~H /\ F e. ~H /\ ( -u S .h G ) e. ~H ) -> ( F .ih ( F +h ( -u S .h G ) ) ) = ( ( F .ih F ) + ( F .ih ( -u S .h G ) ) ) )
20	1, 1, 15, 19	mp3an 1424	. . . 4 |- ( F .ih ( F +h ( -u S .h G ) ) ) = ( ( F .ih F ) + ( F .ih ( -u S .h G ) ) )
21		his5 27943	. . . . . . 7 |- ( ( -u S e. CC /\ F e. ~H /\ G e. ~H ) -> ( F .ih ( -u S .h G ) ) = ( ( * ` -u S ) x. ( F .ih G ) ) )
22	14, 1, 3, 21	mp3an 1424	. . . . . 6 |- ( F .ih ( -u S .h G ) ) = ( ( * ` -u S ) x. ( F .ih G ) )
23	2	cjnegi 13922	. . . . . . 7 |- ( * ` -u S ) = -u ( * ` S )
24	23	oveq1i 6660	. . . . . 6 |- ( ( * ` -u S ) x. ( F .ih G ) ) = ( -u ( * ` S ) x. ( F .ih G ) )
25	22, 24	eqtri 2644	. . . . 5 |- ( F .ih ( -u S .h G ) ) = ( -u ( * ` S ) x. ( F .ih G ) )
26	25	oveq2i 6661	. . . 4 |- ( ( F .ih F ) + ( F .ih ( -u S .h G ) ) ) = ( ( F .ih F ) + ( -u ( * ` S ) x. ( F .ih G ) ) )
27	20, 26	eqtri 2644	. . 3 |- ( F .ih ( F +h ( -u S .h G ) ) ) = ( ( F .ih F ) + ( -u ( * ` S ) x. ( F .ih G ) ) )
28		ax-his3 27941	. . . . 5 |- ( ( -u S e. CC /\ G e. ~H /\ ( F +h ( -u S .h G ) ) e. ~H ) -> ( ( -u S .h G ) .ih ( F +h ( -u S .h G ) ) ) = ( -u S x. ( G .ih ( F +h ( -u S .h G ) ) ) ) )
29	14, 3, 16, 28	mp3an 1424	. . . 4 |- ( ( -u S .h G ) .ih ( F +h ( -u S .h G ) ) ) = ( -u S x. ( G .ih ( F +h ( -u S .h G ) ) ) )
30		his7 27947	. . . . . . 7 |- ( ( G e. ~H /\ F e. ~H /\ ( -u S .h G ) e. ~H ) -> ( G .ih ( F +h ( -u S .h G ) ) ) = ( ( G .ih F ) + ( G .ih ( -u S .h G ) ) ) )
31	3, 1, 15, 30	mp3an 1424	. . . . . 6 |- ( G .ih ( F +h ( -u S .h G ) ) ) = ( ( G .ih F ) + ( G .ih ( -u S .h G ) ) )
32		his5 27943	. . . . . . . 8 |- ( ( -u S e. CC /\ G e. ~H /\ G e. ~H ) -> ( G .ih ( -u S .h G ) ) = ( ( * ` -u S ) x. ( G .ih G ) ) )
33	14, 3, 3, 32	mp3an 1424	. . . . . . 7 |- ( G .ih ( -u S .h G ) ) = ( ( * ` -u S ) x. ( G .ih G ) )
34	33	oveq2i 6661	. . . . . 6 |- ( ( G .ih F ) + ( G .ih ( -u S .h G ) ) ) = ( ( G .ih F ) + ( ( * ` -u S ) x. ( G .ih G ) ) )
35	31, 34	eqtri 2644	. . . . 5 |- ( G .ih ( F +h ( -u S .h G ) ) ) = ( ( G .ih F ) + ( ( * ` -u S ) x. ( G .ih G ) ) )
36	35	oveq2i 6661	. . . 4 |- ( -u S x. ( G .ih ( F +h ( -u S .h G ) ) ) ) = ( -u S x. ( ( G .ih F ) + ( ( * ` -u S ) x. ( G .ih G ) ) ) )
37	3, 1	hicli 27938	. . . . . 6 |- ( G .ih F ) e. CC
38	14	cjcli 13909	. . . . . . 7 |- ( * ` -u S ) e. CC
39	3, 3	hicli 27938	. . . . . . 7 |- ( G .ih G ) e. CC
40	38, 39	mulcli 10045	. . . . . 6 |- ( ( * ` -u S ) x. ( G .ih G ) ) e. CC
41	14, 37, 40	adddii 10050	. . . . 5 |- ( -u S x. ( ( G .ih F ) + ( ( * ` -u S ) x. ( G .ih G ) ) ) ) = ( ( -u S x. ( G .ih F ) ) + ( -u S x. ( ( * ` -u S ) x. ( G .ih G ) ) ) )
42	14, 38, 39	mulassi 10049	. . . . . . 7 |- ( ( -u S x. ( * ` -u S ) ) x. ( G .ih G ) ) = ( -u S x. ( ( * ` -u S ) x. ( G .ih G ) ) )
43	23	oveq2i 6661	. . . . . . . . 9 |- ( -u S x. ( * ` -u S ) ) = ( -u S x. -u ( * ` S ) )
44	2	cjcli 13909	. . . . . . . . . 10 |- ( * ` S ) e. CC
45	2, 44	mul2negi 10478	. . . . . . . . 9 |- ( -u S x. -u ( * ` S ) ) = ( S x. ( * ` S ) )
46	43, 45	eqtri 2644	. . . . . . . 8 |- ( -u S x. ( * ` -u S ) ) = ( S x. ( * ` S ) )
47	46	oveq1i 6660	. . . . . . 7 |- ( ( -u S x. ( * ` -u S ) ) x. ( G .ih G ) ) = ( ( S x. ( * ` S ) ) x. ( G .ih G ) )
48	42, 47	eqtr3i 2646	. . . . . 6 |- ( -u S x. ( ( * ` -u S ) x. ( G .ih G ) ) ) = ( ( S x. ( * ` S ) ) x. ( G .ih G ) )
49	48	oveq2i 6661	. . . . 5 |- ( ( -u S x. ( G .ih F ) ) + ( -u S x. ( ( * ` -u S ) x. ( G .ih G ) ) ) ) = ( ( -u S x. ( G .ih F ) ) + ( ( S x. ( * ` S ) ) x. ( G .ih G ) ) )
50	41, 49	eqtri 2644	. . . 4 |- ( -u S x. ( ( G .ih F ) + ( ( * ` -u S ) x. ( G .ih G ) ) ) ) = ( ( -u S x. ( G .ih F ) ) + ( ( S x. ( * ` S ) ) x. ( G .ih G ) ) )
51	29, 36, 50	3eqtri 2648	. . 3 |- ( ( -u S .h G ) .ih ( F +h ( -u S .h G ) ) ) = ( ( -u S x. ( G .ih F ) ) + ( ( S x. ( * ` S ) ) x. ( G .ih G ) ) )
52	27, 51	oveq12i 6662	. 2 |- ( ( F .ih ( F +h ( -u S .h G ) ) ) + ( ( -u S .h G ) .ih ( F +h ( -u S .h G ) ) ) ) = ( ( ( F .ih F ) + ( -u ( * ` S ) x. ( F .ih G ) ) ) + ( ( -u S x. ( G .ih F ) ) + ( ( S x. ( * ` S ) ) x. ( G .ih G ) ) ) )
53	13, 18, 52	3eqtri 2648	1 |- ( ( F -h ( S .h G ) ) .ih ( F -h ( S .h G ) ) ) = ( ( ( F .ih F ) + ( -u ( * ` S ) x. ( F .ih G ) ) ) + ( ( -u S x. ( G .ih F ) ) + ( ( S x. ( * ` S ) ) x. ( G .ih G ) ) ) )

Syntax hints:   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   -ucneg 10267   *ccj 13836   ~Hchil 27776   +h cva 27777   .h csm 27778   .ih csp 27779   -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-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-hfvadd 27857  ax-hfvmul 27862  ax-hvmulass 27864  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-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-cj 13839  df-re 13840  df-im 13841  df-hvsub 27828

This theorem is referenced by:  normlem1  27967