Theorem normlem9 27975

Description: Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)

Hypotheses

Ref	Expression
normlem8.1	|- A e. ~H
normlem8.2	|- B e. ~H
normlem8.3	|- C e. ~H
normlem8.4	|- D e. ~H

Assertion

Ref	Expression
normlem9	|- ( ( A -h B ) .ih ( C -h D ) ) = ( ( ( A .ih C ) + ( B .ih D ) ) - ( ( A .ih D ) + ( B .ih C ) ) )

Proof of Theorem normlem9

Step	Hyp	Ref	Expression
1		normlem8.1	. . . 4 |- A e. ~H
2		normlem8.2	. . . 4 |- B e. ~H
3	1, 2	hvsubvali 27877	. . 3 |- ( A -h B ) = ( A +h ( -u 1 .h B ) )
4		normlem8.3	. . . 4 |- C e. ~H
5		normlem8.4	. . . 4 |- D e. ~H
6	4, 5	hvsubvali 27877	. . 3 |- ( C -h D ) = ( C +h ( -u 1 .h D ) )
7	3, 6	oveq12i 6662	. 2 |- ( ( A -h B ) .ih ( C -h D ) ) = ( ( A +h ( -u 1 .h B ) ) .ih ( C +h ( -u 1 .h D ) ) )
8		neg1cn 11124	. . . 4 |- -u 1 e. CC
9	8, 2	hvmulcli 27871	. . 3 |- ( -u 1 .h B ) e. ~H
10	8, 5	hvmulcli 27871	. . 3 |- ( -u 1 .h D ) e. ~H
11	1, 9, 4, 10	normlem8 27974	. 2 |- ( ( A +h ( -u 1 .h B ) ) .ih ( C +h ( -u 1 .h D ) ) ) = ( ( ( A .ih C ) + ( ( -u 1 .h B ) .ih ( -u 1 .h D ) ) ) + ( ( A .ih ( -u 1 .h D ) ) + ( ( -u 1 .h B ) .ih C ) ) )
12		ax-his3 27941	. . . . . . 7 |- ( ( -u 1 e. CC /\ B e. ~H /\ ( -u 1 .h D ) e. ~H ) -> ( ( -u 1 .h B ) .ih ( -u 1 .h D ) ) = ( -u 1 x. ( B .ih ( -u 1 .h D ) ) ) )
13	8, 2, 10, 12	mp3an 1424	. . . . . 6 |- ( ( -u 1 .h B ) .ih ( -u 1 .h D ) ) = ( -u 1 x. ( B .ih ( -u 1 .h D ) ) )
14		his5 27943	. . . . . . . 8 |- ( ( -u 1 e. CC /\ B e. ~H /\ D e. ~H ) -> ( B .ih ( -u 1 .h D ) ) = ( ( * ` -u 1 ) x. ( B .ih D ) ) )
15	8, 2, 5, 14	mp3an 1424	. . . . . . 7 |- ( B .ih ( -u 1 .h D ) ) = ( ( * ` -u 1 ) x. ( B .ih D ) )
16	15	oveq2i 6661	. . . . . 6 |- ( -u 1 x. ( B .ih ( -u 1 .h D ) ) ) = ( -u 1 x. ( ( * ` -u 1 ) x. ( B .ih D ) ) )
17		neg1rr 11125	. . . . . . . . . . 11 |- -u 1 e. RR
18		cjre 13879	. . . . . . . . . . 11 |- ( -u 1 e. RR -> ( * ` -u 1 ) = -u 1 )
19	17, 18	ax-mp 5	. . . . . . . . . 10 |- ( * ` -u 1 ) = -u 1
20	19	oveq2i 6661	. . . . . . . . 9 |- ( -u 1 x. ( * ` -u 1 ) ) = ( -u 1 x. -u 1 )
21		ax-1cn 9994	. . . . . . . . . 10 |- 1 e. CC
22	21, 21	mul2negi 10478	. . . . . . . . 9 |- ( -u 1 x. -u 1 ) = ( 1 x. 1 )
23	21	mulid2i 10043	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
24	20, 22, 23	3eqtri 2648	. . . . . . . 8 |- ( -u 1 x. ( * ` -u 1 ) ) = 1
25	24	oveq1i 6660	. . . . . . 7 |- ( ( -u 1 x. ( * ` -u 1 ) ) x. ( B .ih D ) ) = ( 1 x. ( B .ih D ) )
26	8	cjcli 13909	. . . . . . . 8 |- ( * ` -u 1 ) e. CC
27	2, 5	hicli 27938	. . . . . . . 8 |- ( B .ih D ) e. CC
28	8, 26, 27	mulassi 10049	. . . . . . 7 |- ( ( -u 1 x. ( * ` -u 1 ) ) x. ( B .ih D ) ) = ( -u 1 x. ( ( * ` -u 1 ) x. ( B .ih D ) ) )
29	27	mulid2i 10043	. . . . . . 7 |- ( 1 x. ( B .ih D ) ) = ( B .ih D )
30	25, 28, 29	3eqtr3i 2652	. . . . . 6 |- ( -u 1 x. ( ( * ` -u 1 ) x. ( B .ih D ) ) ) = ( B .ih D )
31	13, 16, 30	3eqtri 2648	. . . . 5 |- ( ( -u 1 .h B ) .ih ( -u 1 .h D ) ) = ( B .ih D )
32	31	oveq2i 6661	. . . 4 |- ( ( A .ih C ) + ( ( -u 1 .h B ) .ih ( -u 1 .h D ) ) ) = ( ( A .ih C ) + ( B .ih D ) )
33		his5 27943	. . . . . . . 8 |- ( ( -u 1 e. CC /\ A e. ~H /\ D e. ~H ) -> ( A .ih ( -u 1 .h D ) ) = ( ( * ` -u 1 ) x. ( A .ih D ) ) )
34	8, 1, 5, 33	mp3an 1424	. . . . . . 7 |- ( A .ih ( -u 1 .h D ) ) = ( ( * ` -u 1 ) x. ( A .ih D ) )
35	19	oveq1i 6660	. . . . . . 7 |- ( ( * ` -u 1 ) x. ( A .ih D ) ) = ( -u 1 x. ( A .ih D ) )
36	1, 5	hicli 27938	. . . . . . . 8 |- ( A .ih D ) e. CC
37	36	mulm1i 10475	. . . . . . 7 |- ( -u 1 x. ( A .ih D ) ) = -u ( A .ih D )
38	34, 35, 37	3eqtri 2648	. . . . . 6 |- ( A .ih ( -u 1 .h D ) ) = -u ( A .ih D )
39		ax-his3 27941	. . . . . . . 8 |- ( ( -u 1 e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( -u 1 .h B ) .ih C ) = ( -u 1 x. ( B .ih C ) ) )
40	8, 2, 4, 39	mp3an 1424	. . . . . . 7 |- ( ( -u 1 .h B ) .ih C ) = ( -u 1 x. ( B .ih C ) )
41	2, 4	hicli 27938	. . . . . . . 8 |- ( B .ih C ) e. CC
42	41	mulm1i 10475	. . . . . . 7 |- ( -u 1 x. ( B .ih C ) ) = -u ( B .ih C )
43	40, 42	eqtri 2644	. . . . . 6 |- ( ( -u 1 .h B ) .ih C ) = -u ( B .ih C )
44	38, 43	oveq12i 6662	. . . . 5 |- ( ( A .ih ( -u 1 .h D ) ) + ( ( -u 1 .h B ) .ih C ) ) = ( -u ( A .ih D ) + -u ( B .ih C ) )
45	36, 41	negdii 10365	. . . . 5 |- -u ( ( A .ih D ) + ( B .ih C ) ) = ( -u ( A .ih D ) + -u ( B .ih C ) )
46	44, 45	eqtr4i 2647	. . . 4 |- ( ( A .ih ( -u 1 .h D ) ) + ( ( -u 1 .h B ) .ih C ) ) = -u ( ( A .ih D ) + ( B .ih C ) )
47	32, 46	oveq12i 6662	. . 3 |- ( ( ( A .ih C ) + ( ( -u 1 .h B ) .ih ( -u 1 .h D ) ) ) + ( ( A .ih ( -u 1 .h D ) ) + ( ( -u 1 .h B ) .ih C ) ) ) = ( ( ( A .ih C ) + ( B .ih D ) ) + -u ( ( A .ih D ) + ( B .ih C ) ) )
48	1, 4	hicli 27938	. . . . 5 |- ( A .ih C ) e. CC
49	48, 27	addcli 10044	. . . 4 |- ( ( A .ih C ) + ( B .ih D ) ) e. CC
50	36, 41	addcli 10044	. . . 4 |- ( ( A .ih D ) + ( B .ih C ) ) e. CC
51	49, 50	negsubi 10359	. . 3 |- ( ( ( A .ih C ) + ( B .ih D ) ) + -u ( ( A .ih D ) + ( B .ih C ) ) ) = ( ( ( A .ih C ) + ( B .ih D ) ) - ( ( A .ih D ) + ( B .ih C ) ) )
52	47, 51	eqtri 2644	. 2 |- ( ( ( A .ih C ) + ( ( -u 1 .h B ) .ih ( -u 1 .h D ) ) ) + ( ( A .ih ( -u 1 .h D ) ) + ( ( -u 1 .h B ) .ih C ) ) ) = ( ( ( A .ih C ) + ( B .ih D ) ) - ( ( A .ih D ) + ( B .ih C ) ) )
53	7, 11, 52	3eqtri 2648	1 |- ( ( A -h B ) .ih ( C -h D ) ) = ( ( ( A .ih C ) + ( B .ih D ) ) - ( ( A .ih D ) + ( B .ih C ) ) )

Syntax hints:   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -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-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:  bcseqi  27977  normlem9at  27978  normpari  28011  polid2i  28014