Theorem normlem3 8973

Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.

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)))
normlem3.5	|- A = (G .ih G)
normlem3.6	|- C = (F .ih F)
normlem3.7	|- R e. RR

Assertion

Ref	Expression
normlem3	|- (((A x. (R^2)) + (B x. R)) + C) = (((F .ih F) + (((*` S) x. -uR) x. (F .ih G))) + (((S x. -uR) x. (G .ih F)) + ((R^2) x. (G .ih G))))

Proof of Theorem normlem3

Step	Hyp	Ref	Expression
1		normlem3.6	. . 3 |- C = (F .ih F)
2		normlem3.5	. . . . . . 7 |- A = (G .ih G)
3		normlem1.3	. . . . . . . 8 |- G e. H~
4	3, 3	hicl 8943	. . . . . . 7 |- (G .ih G) e. CC
5	2, 4	eqeltr 1547	. . . . . 6 |- A e. CC
6		normlem3.7	. . . . . . . 8 |- R e. RR
7	6	recn 5326	. . . . . . 7 |- R e. CC
8	7	sqcl 6616	. . . . . 6 |- (R^2) e. CC
9	5, 8	mulcl 5333	. . . . 5 |- (A x. (R^2)) e. CC
10		normlem1.1	. . . . . . . 8 |- S e. CC
11		normlem1.2	. . . . . . . 8 |- F e. H~
12		normlem2.4	. . . . . . . 8 |- B = -u(((*` S) x. (F .ih G)) + (S x. (G .ih F)))
13	10, 11, 3, 12	normlem2 8972	. . . . . . 7 |- B e. RR
14	13	recn 5326	. . . . . 6 |- B e. CC
15	14, 7	mulcl 5333	. . . . 5 |- (B x. R) e. CC
16	9, 15	addcom 5334	. . . 4 |- ((A x. (R^2)) + (B x. R)) = ((B x. R) + (A x. (R^2)))
17	10	cjcl 6768	. . . . . . . . . 10 |- (*` S) e. CC
18	11, 3	hicl 8943	. . . . . . . . . 10 |- (F .ih G) e. CC
19	17, 18	mulcl 5333	. . . . . . . . 9 |- ((*` S) x. (F .ih G)) e. CC
20	3, 11	hicl 8943	. . . . . . . . . 10 |- (G .ih F) e. CC
21	10, 20	mulcl 5333	. . . . . . . . 9 |- (S x. (G .ih F)) e. CC
22	19, 21	addcl 5332	. . . . . . . 8 |- (((*` S) x. (F .ih G)) + (S x. (G .ih F))) e. CC
23	22, 7	mulneg1 5457	. . . . . . 7 |- (-u(((*` S) x. (F .ih G)) + (S x. (G .ih F))) x. R) = -u((((*` S) x. (F .ih G)) + (S x. (G .ih F))) x. R)
24	12	opreq1i 3977	. . . . . . 7 |- (B x. R) = (-u(((*` S) x. (F .ih G)) + (S x. (G .ih F))) x. R)
25	22, 7	mulneg2 5458	. . . . . . 7 |- ((((*` S) x. (F .ih G)) + (S x. (G .ih F))) x. -uR) = -u((((*` S) x. (F .ih G)) + (S x. (G .ih F))) x. R)
26	23, 24, 25	3eqtr4 1508	. . . . . 6 |- (B x. R) = ((((*` S) x. (F .ih G)) + (S x. (G .ih F))) x. -uR)
27	7	negcl 5381	. . . . . . 7 |- -uR e. CC
28	19, 21, 27	adddir 5339	. . . . . 6 |- ((((*` S) x. (F .ih G)) + (S x. (G .ih F))) x. -uR) = ((((*` S) x. (F .ih G)) x. -uR) + ((S x. (G .ih F)) x. -uR))
29	17, 18, 27	mul23 5436	. . . . . . 7 |- (((*` S) x. (F .ih G)) x. -uR) = (((*` S) x. -uR) x. (F .ih G))
30	10, 20, 27	mul23 5436	. . . . . . 7 |- ((S x. (G .ih F)) x. -uR) = ((S x. -uR) x. (G .ih F))
31	29, 30	opreq12i 3979	. . . . . 6 |- ((((*` S) x. (F .ih G)) x. -uR) + ((S x. (G .ih F)) x. -uR)) = ((((*` S) x. -uR) x. (F .ih G)) + ((S x. -uR) x. (G .ih F)))
32	26, 28, 31	3eqtr 1502	. . . . 5 |- (B x. R) = ((((*` S) x. -uR) x. (F .ih G)) + ((S x. -uR) x. (G .ih F)))
33	8, 5	mulcom 5335	. . . . . 6 |- ((R^2) x. A) = (A x. (R^2))
34	2	opreq2i 3978	. . . . . 6 |- ((R^2) x. A) = ((R^2) x. (G .ih G))
35	33, 34	eqtr3 1500	. . . . 5 |- (A x. (R^2)) = ((R^2) x. (G .ih G))
36	32, 35	opreq12i 3979	. . . 4 |- ((B x. R) + (A x. (R^2))) = (((((*` S) x. -uR) x. (F .ih G)) + ((S x. -uR) x. (G .ih F))) + ((R^2) x. (G .ih G)))
37	17, 27	mulcl 5333	. . . . . 6 |- ((*` S) x. -uR) e. CC
38	37, 18	mulcl 5333	. . . . 5 |- (((*` S) x. -uR) x. (F .ih G)) e. CC
39	10, 27	mulcl 5333	. . . . . 6 |- (S x. -uR) e. CC
40	39, 20	mulcl 5333	. . . . 5 |- ((S x. -uR) x. (G .ih F)) e. CC
41	8, 4	mulcl 5333	. . . . 5 |- ((R^2) x. (G .ih G)) e. CC
42	38, 40, 41	addass 5336	. . . 4 |- (((((*` S) x. -uR) x. (F .ih G)) + ((S x. -uR) x. (G .ih F))) + ((R^2) x. (G .ih G))) = ((((*` S) x. -uR) x. (F .ih G)) + (((S x. -uR) x. (G .ih F)) + ((R^2) x. (G .ih G))))
43	16, 36, 42	3eqtr 1502	. . 3 |- ((A x. (R^2)) + (B x. R)) = ((((*` S) x. -uR) x. (F .ih G)) + (((S x. -uR) x. (G .ih F)) + ((R^2) x. (G .ih G))))
44	1, 43	opreq12i 3979	. 2 |- (C + ((A x. (R^2)) + (B x. R))) = ((F .ih F) + ((((*` S) x. -uR) x. (F .ih G)) + (((S x. -uR) x. (G .ih F)) + ((R^2) x. (G .ih G)))))
45	9, 15	addcl 5332	. . 3 |- ((A x. (R^2)) + (B x. R)) e. CC
46	11, 11	hicl 8943	. . . 4 |- (F .ih F) e. CC
47	1, 46	eqeltr 1547	. . 3 |- C e. CC
48	45, 47	addcom 5334	. 2 |- (((A x. (R^2)) + (B x. R)) + C) = (C + ((A x. (R^2)) + (B x. R)))
49	40, 41	addcl 5332	. . 3 |- (((S x. -uR) x. (G .ih F)) + ((R^2) x. (G .ih G))) e. CC
50	46, 38, 49	addass 5336	. 2 |- (((F .ih F) + (((*` S) x. -uR) x. (F .ih G))) + (((S x. -uR) x. (G .ih F)) + ((R^2) x. (G .ih G)))) = ((F .ih F) + ((((*` S) x. -uR) x. (F .ih G)) + (((S x. -uR) x. (G .ih F)) + ((R^2) x. (G .ih G)))))
51	44, 48, 50	3eqtr4 1508	1 |- (((A x. (R^2)) + (B x. R)) + C) = (((F .ih F) + (((*` S) x. -uR) x. (F .ih G))) + (((S x. -uR) x. (G .ih F)) + ((R^2) x. (G .ih G))))

Syntax hints:   = wceq 958   e. wcel 960  ` cfv 3188  (class class class)co 3969  CCcc 5244  RRcr 5245   + caddc 5249   x. cmul 5251  -ucneg 5305  2c2 5963  ^cexp 6569  *ccj 6750  H~chil 8783   .ih csp 8788

This theorem is referenced by:  normlem4 8974

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-hfi 8941  ax-his1 8944

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-n0 6102  df-z 6138  df-seq1 6309  df-exp 6570  df-re 6752  df-im 6753  df-cj 6754

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