Theorem normlem2 27968

Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
normlem1.1	|- S e. CC
normlem1.2	|- F e. ~H
normlem1.3	|- G e. ~H
normlem2.4	|- B = -u ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) )

Assertion

Ref	Expression
normlem2	|- B e. RR

Proof of Theorem normlem2

Step	Hyp	Ref	Expression
1		normlem2.4	. 2 |- B = -u ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) )
2		normlem1.1	. . . . . . . . 9 |- S e. CC
3	2	cjcli 13909	. . . . . . . 8 |- ( * ` S ) e. CC
4		normlem1.2	. . . . . . . . 9 |- F e. ~H
5		normlem1.3	. . . . . . . . 9 |- G e. ~H
6	4, 5	hicli 27938	. . . . . . . 8 |- ( F .ih G ) e. CC
7	3, 6	mulcli 10045	. . . . . . 7 |- ( ( * ` S ) x. ( F .ih G ) ) e. CC
8	5, 4	hicli 27938	. . . . . . . 8 |- ( G .ih F ) e. CC
9	2, 8	mulcli 10045	. . . . . . 7 |- ( S x. ( G .ih F ) ) e. CC
10	7, 9	cjaddi 13928	. . . . . 6 |- ( * ` ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) ) ) = ( ( * ` ( ( * ` S ) x. ( F .ih G ) ) ) + ( * ` ( S x. ( G .ih F ) ) ) )
11	2	cjcji 13911	. . . . . . . . . 10 |- ( * ` ( * ` S ) ) = S
12	11	eqcomi 2631	. . . . . . . . 9 |- S = ( * ` ( * ` S ) )
13	5, 4	his1i 27957	. . . . . . . . 9 |- ( G .ih F ) = ( * ` ( F .ih G ) )
14	12, 13	oveq12i 6662	. . . . . . . 8 |- ( S x. ( G .ih F ) ) = ( ( * ` ( * ` S ) ) x. ( * ` ( F .ih G ) ) )
15	3, 6	cjmuli 13929	. . . . . . . 8 |- ( * ` ( ( * ` S ) x. ( F .ih G ) ) ) = ( ( * ` ( * ` S ) ) x. ( * ` ( F .ih G ) ) )
16	14, 15	eqtr4i 2647	. . . . . . 7 |- ( S x. ( G .ih F ) ) = ( * ` ( ( * ` S ) x. ( F .ih G ) ) )
17	4, 5	his1i 27957	. . . . . . . . 9 |- ( F .ih G ) = ( * ` ( G .ih F ) )
18	17	oveq2i 6661	. . . . . . . 8 |- ( ( * ` S ) x. ( F .ih G ) ) = ( ( * ` S ) x. ( * ` ( G .ih F ) ) )
19	2, 8	cjmuli 13929	. . . . . . . 8 |- ( * ` ( S x. ( G .ih F ) ) ) = ( ( * ` S ) x. ( * ` ( G .ih F ) ) )
20	18, 19	eqtr4i 2647	. . . . . . 7 |- ( ( * ` S ) x. ( F .ih G ) ) = ( * ` ( S x. ( G .ih F ) ) )
21	16, 20	oveq12i 6662	. . . . . 6 |- ( ( S x. ( G .ih F ) ) + ( ( * ` S ) x. ( F .ih G ) ) ) = ( ( * ` ( ( * ` S ) x. ( F .ih G ) ) ) + ( * ` ( S x. ( G .ih F ) ) ) )
22	10, 21	eqtr4i 2647	. . . . 5 |- ( * ` ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) ) ) = ( ( S x. ( G .ih F ) ) + ( ( * ` S ) x. ( F .ih G ) ) )
23	7, 9	addcomi 10227	. . . . 5 |- ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) ) = ( ( S x. ( G .ih F ) ) + ( ( * ` S ) x. ( F .ih G ) ) )
24	22, 23	eqtr4i 2647	. . . 4 |- ( * ` ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) ) ) = ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) )
25	7, 9	addcli 10044	. . . . 5 |- ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) ) e. CC
26	25	cjrebi 13914	. . . 4 |- ( ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) ) e. RR <-> ( * ` ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) ) ) = ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) ) )
27	24, 26	mpbir 221	. . 3 |- ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) ) e. RR
28	27	renegcli 10342	. 2 |- -u ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) ) e. RR
29	1, 28	eqeltri 2697	1 |- B e. RR

Syntax hints:   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   + caddc 9939   x. cmul 9941   -ucneg 10267   *ccj 13836   ~Hchil 27776   .ih csp 27779

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-hfi 27936  ax-his1 27939

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

This theorem is referenced by:  normlem3  27969  normlem6  27972  normlem7  27973  norm-ii-i  27994